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arXiv:1807.03206 · Physical Review Letters

Black-Box Quantum State Preparation without Arithmetic

Yuval R. Sanders, Guang Hao Low, Artur Scherer, Dominic William Berry

Published Jan 16, 2019• 109 citations
arXivHTMLPDF

Abstract

Black-box quantum state preparation is an important subroutine in many quantum algorithms. The standard approach requires the quantum computer to do arithmetic, which is a key contributor to the complexity. Here we present a new algorithm that avoids arithmetic. We thereby reduce the number of gates by a factor of 286-374 over the best prior work for realistic precision; the improvement factor increases with the precision. As quantum state preparation is a crucial subroutine in many approaches to simulating physics on a quantum computer, our new method brings useful quantum simulation closer to reality.

DOI
10.1103/physrevlett.122.020502

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Black-box quantum state preparation is an important subroutine in many quantum algorithms. The standard approach requires the quantum computer to do arithmetic, which is a key contributor to the complexity. Here we present a new algorithm that avoids arithmetic. We thereby reduce the number of gates by a factor of 286-374 over the best prior work for realistic precision; the improvement factor increases with the precision. As quantum state preparation is a crucial subroutine in many approaches to simulating physics on a quantum computer, our new method brings useful quantum simulation closer to reality.
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Quantum state preparation is an important class of quantum algorithms that is employed as a black-box subroutine in many algorithms, or used by itself to generate arbitrary probability distributions. We present a novel state preparation method that utilizes less quantum computing resource than the existing methods. Two variants of the algorithm with different emphases are introduced. One variant uses fewer qubits and no controlled gates, while the other variant potentially requires fewer gates overall. A general analysis is given to estimate the number of qubits necessary to achieve a desired precision in the amplitudes of the computational basis states. The validity of the algorithm is demonstrated using a prototypical problem of generating Ising model spin configurations according to its Boltzmann distribution.
The preparation of nnn-qubit quantum states is a cross-cutting subroutine for many quantum algorithms, and the effort to reduce its circuit complexity is a significant challenge. In the literature, the quantum state preparation algorithm by Sun et al. is known to be optimally bounded, defining the asymptotically optimal width-depth trade-off bounds with and without ancillary qubits. In this work, a simpler algebraic decomposition is proposed to separate the preparation of the real part of the desired state from the complex one, resulting in a reduction in terms of circuit depth, total gates, and CNOT count when mmm ancillary qubits are available. The reduction in complexity is due to the use of a single operator ΛΛΛ for each uniformly controlled gate, instead of the three in the original decomposition. Using the PennyLane library, this new algorithm for state preparation has been implemented and tested in a simulated environment for both dense and sparse quantum states, including those that are random and of physical interest. Furthermore, its performance has been compared with that of Möttönen et al.'s algorithm, which is a de facto standard for preparing quantum states in cases where no ancillary qubits are used, highlighting interesting lines of development.
Black-box quantum-state preparation is a variant of quantum-state preparation where we want to construct an nnn-qubit state ∣ψc⟩∝∑xc(x)∣x⟩|ψ_c\rangle \propto \sum_x c(x) |x\rangle∣ψc​⟩∝∑x​c(x)∣x⟩ with the amplitudes c(x)c(x)c(x) given as a (quantum) oracle. This variant is particularly useful when the quantum state has a short and simple classical description. We use recent techniques, namely quantum signal processing (QSP) and quantum singular value transform (QSVT), to construct a new algorithm that prepares ∣ψc⟩|ψ_c\rangle∣ψc​⟩ without the need to carry out coherent arithmetic. We then compare our result with current state-of-the-art algorithms, showing that a QSVT-based approach achieves comparable results.
Abstract Black-box quantum state preparation is a fundamental building block for many higher-level quantum algorithms. The basic task of black-box state preparation is to transduce the data encoded as computational basis of quantum state into the amplitude. In the present work, we address the problem of transducing the reciprocal of the data, not the data itself into the amplitude, which is called the inverse-coefficient problem. This algorithm can be used directly as a subroutine in the matrix inversion algorithms. Furthermore, we extend this approach to address the more general nonlinear-coefficient problem in black-box state preparation. Our algorithm is based on the technique of inequality test. It can greatly relieve the need to do quantum arithmetic and the error is only resulted from the truncated error of binary string. The present algorithms enrich the algorithm library of black-box quantum state preparation and will be useful ingredients of quantum algorithm to implement non-linear quantum state transformations.
The initialization of quantum states or Quantum State Preparation (QSP) is a basic subroutine in quantum algorithms. In the worst case, general QSP algorithms are expensive due to the application of multi-controlled gates required to build the quantum state. Here, we propose an algorithm that detects whether a given quantum state can be factored into substates, increasing the efficiency of compiling the QSP circuit when we initialize states with some level of unentanglement. The simplification is done by eliminating controls of quantum multiplexers, significantly reducing circuit depth and the number of CNOT gates with a better execution and compilation time than the previous QSP algorithms. Considering efficiency in terms of depth and number of CNOT gates, our method is competitive with the methods in the literature. However, when it comes to run-time and compilation efficiency, our result is significantly better, and the experiments show that by increasing the number of qubits, the gap between the temporal efficiency of the methods increases.