Linear combination of unitaries (LCU for short) is one of the most important techniques in designing quantum algorithms. In this paper, we propose a new quantum algorithm in three different forms to achieve LCU. Different from previous algorithms [Childs-linear-system,Clader,Long11], the complexity now only depends on the number of the unitaries and the precision. So it will play more important role in the design of quantum algorithms when the number of unitaries is small, such as quantum iteration algorithms. %Since in iteration algorithms, $m$ often refers to the iteration steps, which is small if the iteration algorithms are efficient. Moreover, as an application of the new LCU, three new quantum algorithms to the searching problem will be proposed, which will provide us new insights into Grover's searching algorithm. We also show that the problem of LCU is closely related to the problem of if we can efficiently implement $U^t$ for $0
Linear combination of unitaries (LCU for short) is one of the most important techniques in designing quantum algorithms. In this paper, we propose a new quantum algorithm in three different forms to achieve LCU. Different from previous algorithms [Childs-linear-system,Clader,Long11], the complexity now only depends on the number of the unitaries and the precision. So it will play more important role in the design of quantum algorithms when the number of unitaries is small, such as quantum iteration algorithms. %Since in iteration algorithms, m often refers to the iteration steps, which is small if the iteration algorithms are efficient. Moreover, as an application of the new LCU, three new quantum algorithms to the searching problem will be proposed, which will provide us new insights into Grover's searching algorithm. We also show that the problem of LCU is closely related to the problem of if we can efficiently implement Ut for $0
We develop three new methods to implement any Linear Combination of Unitaries (LCU), a powerful quantum algorithmic tool with diverse applications. While the standard LCU procedure requires several ancilla qubits and sophisticated multi-qubit controlled operations, our methods consume significantly fewer quantum resources. The first method ( Single-Ancilla LCU ) estimates expectation values of observables with respect to any quantum state prepared by an LCU procedure while requiring only a single ancilla qubit, and no multi-qubit controlled operations. The second approach ( Analog LCU ) is a simple, physically motivated, continuous-time analogue of LCU, tailored to hybrid qubit-qumode systems. The third method ( Ancilla-free LCU ) requires no ancilla qubit at all and is useful when we are interested in the projection of a quantum state (prepared by the LCU procedure) in some subspace of interest. We apply the first two techniques to develop new quantum algorithms for a wide range of practical problems, ranging from Hamiltonian simulation, ground state preparation and property estimation, and quantum linear systems. Remarkably, despite consuming fewer quantum resources they retain a provable quantum advantage. The third technique allows us to connect discrete and continuous-time quantum walks with their classical counterparts. It also unifies the recently developed optimal quantum spatial search algorithms in both these frameworks, and leads to the development of new ones that require fewer ancilla qubits. Overall, our results are quite generic and can be readily applied to other problems, even beyond those considered here.
Linear combination of unitaries (LCU) is a fundamental primitive in quantum algorithms, whose cost is typically governed by the most expensive unitary appearing in the combination. We develop a transducer-based LCU framework that reduces this worst-case dependence to a weighted average query complexity, when the constituent unitaries share access to a common set of primitive oracles.
Consider A=∑jcjUj, where cj>0 and each unitary Uj can be implemented using Cj primitive queries. Given an upper bound a≥∥A∥, our algorithm implements a block-encoding of A/α with rescaling factor α=O(a), using O(Cmax+Cλ/a) primitive queries, where λ=∑jcj, Cmax=maxjCj, and C=∑jcjCj/λ. By comparison, the standard LCU construction requires O(Cmaxλ/a) primitive queries. The improvement can therefore be substantial when costly unitaries have small weights and λ/a is large. As applications, we obtain improved block-encodings of sparse matrices, leading to quantum algorithms for sparse Hamiltonian simulation and quantum linear systems with near-optimal query complexity up to polylogarithmic factors in all relevant parameters.
Our main technique is the transducer framework developed by Belovs, Jeffery, and Yolcu [Quantum, 8:1444 (2024)]. Here we develop a complementary operator-level theory tailored to block-encodings. We identify the resolvent norm K as a key complexity measure for transducer implementation besides the existing catalyst complexity. We show that the transducer action can be converted into an ε-approximate block-encoding using only O(Klog(1/ε)) queries to the transducer, improving the precision dependence from polynomial to logarithmic.
Solving systems of linear equations is one of the most important primitives in many different areas, including in optimization, simulation, and machine learning. Quantum algorithms for solving linear systems have the potential to provide a quantum advantage for these problems. In this work, we recall the Chebyshev iterative method and the corresponding optimal polynomial approximation of the inverse. We show that the Chebyshev iteration polynomial can be efficiently evaluated both using quantum singular value transformation (QSVT) as well as linear combination of unitaries (LCU). We achieve this by bounding the 1-norm of the coefficients of the polynomial expressed in the Chebyshev basis. This leads to a considerable constant-factor improvement in the runtime of quantum linear system solvers that are based on LCU or QSVT (or, conversely, a several orders of magnitude smaller error with the same runtime/circuit depth).
Arnoldi method and conjugate gradient method are important classical iteration methods in solving linear systems and estimating eigenvalues. Their efficiency often affected by the high dimension of the space, where quantum computer can play a role in. In this work, we establish their corresponding quantum algorithms. To achieve high efficiency, a new method about linear combination of quantum states will be proposed. The final complexity of quantum Arnoldi iteration method is O(m3+log(m/ε)(logn)2/ε4) and the final complexity of quantum conjugate gradient iteration method is O(m1+logm/ε(logn)2κ/ε), where ε is precision parameter, m is the iteration steps, n is the dimension of space and κ is the condition number of the coefficient matrix of the linear system the conjugate gradient method works on. Compared with the classical methods, whose complexity are O(mn2+m2n) and O(mn2) respectively, these two quantum algorithms provide us more efficient methods to solve linear systems and to compute eigenvalues and eigenvectors of general matrices. Different from the work \cite{rebentros}, the complexity here is almost polynomial in the iteration steps. Also this work is more general than the iteration method considered in \cite{kerenidis}.
Quantum computing exploits quantum mechanical properties to perform computations. It enables quantum parallelism and provides much more powerful data processing capabilities than classical computers. With a quantum computer, one can exponentially accelerate quantum system simulation and accelerate some important classical algorithms. Traditional quantum algorithms use unitary evolution to process information. A quantum computing process is the product of a series of unitary operators. In 1994, Shor invented a quantum prime factorization algorithm that exponentially accelerates the factorization of an integer. In 1996, Grover proposed a quantum search algorithm that accelerates the search of an unsorted database using square-root steps. Afterwards, the development of quantum algorithms slowed down, leading to a subsequent query by Shor in 2003 regarding why no more significant quantum algorithms have been found. Since 2009, many important new quantum algorithms have been proposed, such as a quantum algorithm for solving linear equations with the capability of exponential acceleration, a quantum algorithm for sparse Hamiltonian simulation using linear combinations of unitary operators, and a novel algorithm for Hamiltonian simulation, which provides exponential improvements in precision. In this paper, we first describe the basic principles of quantum computation. We then describe the Shor algorithm and Grover/Long search algorithm. These quantum algorithms are the general quantum algorithms that use unitary operations in their computational processes. Next, we introduce the basic principles of duality quantum computing, which was proposed in 2002. In contrast to traditional quantum computing, duality quantum computing allows the use of linear combinations of unitary operators in the computation process, meaning the multiplication, division, addition, and subtraction of unitary operators are all possible in duality quantum computing. Thus, duality quantum computing provides more flexibility for constructing quantum algorithms and the techniques used in classical algorithm design can be directly used to construct quantum algorithms. We then review recent work regarding newer quantum algorithms that enable the linear combination of unitary operators, which are actually duality quantum algorithms. Additionally, a duality quantum simulation algorithm for open quantum systems is introduced. This algorithm not only reduces computational complexity, but also improves accuracy exponentially. Finally, a summary and the future prospects of quantum algorithms are provided.
In order to improve the searching efficiency of the Grover algorithm,a new quantum search algorithm was presented based on the Grover algorithm.This quantum search algorithm makes full use of the structural information of states,and spin quantum gates are designed in light of different initialized quantum states and searched ground states,then the new quantum gates are used to spin iteration.The research shows that compared with the Grover algorithm,the presented algorithm decreases iteration times.The feasibility and validity of this algorithm are verified through practical examples.