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arXiv:2009.10709 · Quantum

Fast Black-Box Quantum State Preparation

Johannes Bausch

Published Aug 4, 2022• 27 citations

Abstract

Quantum state preparation is an important ingredient for other higher-level quantum algorithms, such as Hamiltonian simulation, or for loading distributions into a quantum device to be used e.g. in the context of optimization tasks such as machine learning. Starting with a generic "black box" method devised by Grover in 2000, which employs amplitude amplification to load coefficients calculated by an oracle, there has been a long series of results and improvements with various additional conditions on the amplitudes to be loaded, culminating in Sanders et al.'s work which avoids almost all arithmetic during the preparation stage.In this work, we construct an optimized black box state loading scheme with which various important sets of coefficients can be loaded significantly faster than in O(N) rounds of amplitude amplification, up to only O(1) many. We achieve this with two variants of our algorithm. The first employs a modification of the oracle from Sanders et al., which requires fewer ancillas ( log2⁡g vs g+2 in the bit precision g ), and fewer non-Clifford operations per amplitude amplification round within the context of our algorithm. The second utilizes the same oracle, but at slightly increased cost in terms of ancillas ( g+log2⁡g ) and non-Clifford operations per amplification round. As the number of amplitude amplification rounds enters as multiplicative factor, our black box state loading scheme yields an up to exponential speedup as compared to prior methods. This speedup translates beyond the black box case.

DOI: 10.22331/q-2022-08-04-773

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Quantum state preparation is an important ingredient for other higher-level quantum algorithms, such as Hamiltonian simulation, or for loading distributions into a quantum device to be used e.g. in the context of optimization tasks such as machine learning. Starting with a generic "black box" method devised by Grover in 2000, which employs amplitude amplification to load coefficients calculated by an oracle, there has been a long series of results and improvements with various additional conditions on the amplitudes to be loaded, culminating in Sanders et al.'s work which avoids almost all arithmetic during the preparation stage.In this work, we construct an optimized black box state loading scheme with which various important sets of coefficients can be loaded significantly faster than in O(N) rounds of amplitude amplification, up to only O(1) many. We achieve this with two variants of our algorithm. The first employs a modification of the oracle from Sanders et al., which requires fewer ancillas ( log2⁡g vs g+2 in the bit precision g ), and fewer non-Clifford operations per amplitude amplification round within the context of our algorithm. The second utilizes the same oracle, but at slightly increased cost in terms of ancillas ( g+log2⁡g ) and non-Clifford operations per amplification round. As the number of amplitude amplification rounds enters as multiplicative factor, our black box state loading scheme yields an up to exponential speedup as compared to prior methods. This speedup translates beyond the black box case.

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.

In this paper, we present an algorithm for preparing quantum states of the form

\sum_{i=0}^{n-1}\alpha_{i}|i\rangle

, where the coefficients

\alpha_{i}

are specified by a quantum oracle. Our method achieves this task twice as fast as the best existing algorithm known to the authors. Such state preparation is essential for quantum algorithms that process large classical inputs, including matrix inversion and linear system solvers. The standard approach relies on amplitude amplification, a process that may require multiple, time-consuming oracle queries. Consequently, reducing the number of queries and thereby the overall time complexity can lead to significant performance improvements in practice.

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.

Black-box quantum-state preparation is a variant of quantum-state preparation where we want to construct an

n

-qubit state

|ψ_c\rangle \propto \sum_x c(x) |x\rangle

with the amplitudes

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

|ψ_c\rangle

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.

We describe a simple quantum algorithm for preparing

K

copies of an

N

-dimensional quantum state whose amplitudes are given by a quantum oracle. Our result extends a previous work of Grover, who showed how to prepare one copy in time

O(\sqrt{N})

. In comparison with the naive

O(K\sqrt{N})

solution obtained by repeating this procedure

K

times, our algorithm achieves the optimal running time of $\mathrm{\ensuremath{\Theta}}(\sqrt{KN})$. Our technique uses a refinement of the quantum rejection sampling method employed by Grover. As a direct application, we obtain a similar speedup for obtaining

K

independent samples from a distribution whose probability vector is given by a quantum oracle.

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.