Quantum algorithm involves the manipulation of amplitudes and computational basis, of which manipulating basis is largely a quantum analogue of classical computing that is always a major contributor to the complexity. In order to make full use of quantum mechanical speedup, more transformation should be implemented on amplitudes. Here we propose the notion of quantum amplitude arithmetic (QAA) that intent to evolve the quantum state by performing arithmetic operations on amplitude. Based on the basic design of multiplication and addition operations, QAA can be applied to solve the black-box quantum state preparation problem and the quantum linear system problem with fairly low complexity, and evaluate nonlinear functions on amplitudes directly. QAA is expected to find applications in a variety of quantum algorithms.
Quantum algorithm involves the manipulation of amplitudes and computational basis, of which manipulating basis is largely a quantum analogue of classical computing that is always a major contributor to the complexity. In order to make full use of quantum mechanical speedup, more transformation should be implemented on amplitudes. Here we propose the notion of quantum amplitude arithmetic (QAA) that intent to evolve the quantum state by performing arithmetic operations on amplitude. Based on the basic design of multiplication and addition operations, QAA can be applied to solve the black-box quantum state preparation problem and the quantum linear system problem with fairly low complexity, and evaluate nonlinear functions on amplitudes directly. QAA is expected to find applications in a variety of quantum algorithms.
Quantum algorithms manipulate the amplitudes of quantum states to find solutions to computational problems. In this work, we present a framework for applying a general class of non-linear functions to the amplitudes of quantum states, with up-to an exponential improvement over the previous work. Our framework accepts a state preparation unitary (or block-encoding), specified as a quantum circuit, defining an N-dimensional quantum state. We then construct a diagonal block-encoding of the amplitudes of the quantum state, building on and simplifying previous work. Techniques from the QSVT literature are then used to process this block-encoding. The source of our exponential speedup comes from the quantum analog of importance sampling. We then derive new error-bounds relevant for end-to-end applications, giving the error in terms of ℓ2-norm error. We demonstrate the power of this framework with four key applications. First, our algorithm can apply the important function tanh(x) to the amplitudes of an arbitrary quantum state with at most an ℓ2-norm error of ε, with worst-case query complexity of O(log(N/ε)), in comparison to the O(Nlog(N/ε)) of prior work. Second, we present an algorithm solving a new formulation of maximum finding in the unitary input model. Third, we prove efficient end-to-end complexities in applying a number of common non-linear functions to arbitrary quantum states. Finally, we generalize and unify existing quantum arithmetic-free state-preparation techniques. Our work provides an important and efficient algorithmic building block with potentially numerous applications in areas such as optimization, state preparation, quantum chemistry, and machine learning.
The Quantum Adiabatic Algorithm (QAA) has been proposed as a mechanism for efficiently solving optimization problems on a quantum computer. Since adiabatic computation is analog in nature and does not require the design and use of quantum gates, it can be thought of as a simpler and perhaps more profound method for performing quantum computations that might also be easier to implement experimentally. While these features have generated substantial research in QAA, to date there is still a lack of solid evidence that the algorithm can outperform classical optimization algorithms.
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.
This paper introduces a comprehensive quantum calculator using Qiskit that integrates quantum arithmetic and logic functions into a single calculator. The calculator relies on quantum Fourier transforms (QFT), controlled-phase gates, and quantum modular arithmetic to perform basic arithmetic operations such as addition, subtraction, and multiplication. These arithmetic techniques used quantum parallelism and interference to perform calculations more efficiently, particularly for tasks involving large numerical inputs. Numerical values are stored in qubit registers and processed using QFT-based circuits, which enable efficient and reversible mathematical computations. In addition to arithmetic operations, the calculator also implements quantum logic algorithms, including Grover’s Search and the Deutsch-Jozsa algorithm. Grover’s algorithm is an unstructured search algorithm that operates by constructing an oracle and iteratively amplifying the probability amplitude of the target solution. This amplitude amplification significantly increases the likelihood of measuring the desired outcome. Due to this property, Grover’s algorithm has practical relevance in applications such as database searching and solving combinatorial optimization problems. The Deutsch-Jozsa algorithm determines whether or not a certain Boolean function is constant or balanced and offers an exponential speed up compared to classical methods for this class of problem. A key fundamental function of the calculator is the ability to generate hybrid quantum circuits that support both arithmetic and logical functions. That allows the exploration of simple as well as complex problem-solving strategies that take advantage of both quantum arithmetic and decision-making algorithms. For instance, combining Grover’s amplification with function characterizations from the Deutsch-Jozsa algorithm provides not only interesting circuit configurations but also adds value to the possibilities of the algorithms. This research offers a modular idea and allows exploring the fundamental principles of quantum computing generally speaking and will be a valuable resource for educational as well as research purposes in the rapidly growing field of quantum algorithm development.
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 achieve. In this paper, we define a task called 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. In this paper, we provide a promising way to introduce the highly complex nonlinearity of the quantum states, which is essentially missing in quantum mechanics. Published by the American Physical Society 2024
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 ( log2g 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+log2g ) 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.