Abstract Quantum algorithms of factoring problem have been paid more and more attention since Shor’s algorithm was proposed. Combining the current quantum computer hardware level and integer factorization quantum algorithms, this paper proposes a “classical + quantum” hybrid solution. This scheme first adopts classical methods and corresponding rules in the preprocessing step to simplify the nonlinear equations, which is used to reduce the number of qubits needed for the cost Hamiltonian. Then the variational quantum algorithm is used to find the approximate ground state of the cost Hamiltonian, which encodes the solution of the nonlinear equations. The program was verified on the IBM QX4 quantum machine, and the results showed that this hybrid solution can effectively reduce the quantum resources required to solve the nonlinear equations.
Abstract Quantum algorithms of factoring problem have been paid more and more attention since Shor’s algorithm was proposed. Combining the current quantum computer hardware level and integer factorization quantum algorithms, this paper proposes a “classical + quantum” hybrid solution. This scheme first adopts classical methods and corresponding rules in the preprocessing step to simplify the nonlinear equations, which is used to reduce the number of qubits needed for the cost Hamiltonian. Then the variational quantum algorithm is used to find the approximate ground state of the cost Hamiltonian, which encodes the solution of the nonlinear equations. The program was verified on the IBM QX4 quantum machine, and the results showed that this hybrid solution can effectively reduce the quantum resources required to solve the nonlinear equations.
The content of this poster is a description of the implementation of two well-known quantum integer factorization methods and, in addition, a proposal of a new, third way along with some examples of its use. For all three methods, the measured results of their circuits will be shown.The first method is Shor's algorithm, where a quantum circuit is used only to find the period of a specially assembled modular exponentiating function. The rest of the algorithm is preprocessing and postprocessing that can be done on a classical computer, so one could say that this is a hybrid algorithm. New results of the successful factorization of the number 119, performed by this algorithm on a real 7-qubit quantum computer, will be presented. It is worth mentioning here that the highest number successfully factorized by this algorithm on a real quantum computer so far was 35.The second method is variational quantum factorization (VQF). In this method, preprocessing builds a system of equations based on the classical binary multiplication procedure of two numbers, resulting in a factorized number. This system of equations is then used to form a Hamiltonian in such a way that its ground state corresponds to the solution of this equation system. Only now the hybrid quantum optimization comes into play to iteratively find this ground state. The advantage over Shor's algorithm is that this method does not require a fault-tolerant quantum computer, but on the other hand it approaches the solution iteratively using a classical computer, which reduces the potential quantum advantage.The new, third method employs Grover's search algorithm in an unusual way to create a quantum factorization circuit. A key part of this approach is the quantum implementation of a multiplying oracle. The main advantage of this method over the previous ones is that it can be used to assemble a relatively simple universal factorization circuit. Moreover, there is almost no preprocessing and postprocessing required. This method is still under development, but this poster will present the first results.
In recent years, as datasets have grown larger, quantum computing has shown potential in addressing the rapidly increasing costs of data processing due to its superposition and entanglement capabilities. Limited by quantum hardware and the number of controllable qubits, variational quantum algorithms (VQAs) are algorithms suitable for NISQ devices, capable of computation on medium-scale noisy devices. VQAs are hybrid classical-quantum algorithms that use parameterized quantum circuits to prepare parameterized quantum states and employ classical optimizers to minimize objective functions, enabling efficient solutions for complex problems. Classical optimizers reduce the stringent requirements on quantum hardware and can achieve quantum advantage on current devices. This paper systematically elaborates on the core foundational theories of quantum computing and neural networks, with a focus on reviewing the research and application status of VQAs in three main scenarios: machine learning (supervised, unsupervised, and reinforcement learning), solving systems of equations (linear and nonlinear), and network attack detection. It analyzes the current technical bottlenecks in this field and provides prospects for future development, offering a reference for further research on variational quantum algorithms.
The road to computing on quantum devices has been accelerated by the promises that come from using Shor's algorithm to reduce the complexity of prime factorization. However, this promise hast not yet been realized due to noisy qubits and lack of robust error correction schemes. Here we explore a promising, alternative method for prime factorization that uses well-established techniques from variational imaginary time evolution. We create a Hamiltonian whose ground state encodes the solution to the problem and use variational techniques to evolve a state iteratively towards these prime factors. We show that the number of circuits evaluated in each iteration scales as [Formula: see text], where n is the bit-length of the number to be factorized and d is the depth of the circuit. We use a single layer of entangling gates to factorize 36 numbers represented using 7, 8, and 9-qubit Hamiltonians. We also verify the method's performance by implementing it on the IBMQ Lima hardware to factorize 55, 65, 77 and 91 which are greater than the largest number (21) to have been factorized on IBMQ hardware.
Quantum algorithms have been developed for efficiently solving linear algebra tasks. However, they generally require deep circuits and hence universal fault-tolerant quantum computers. In this work, we propose variational algorithms for linear algebra tasks that are compatible with noisy intermediate-scale quantum devices. We show that the solutions of linear systems of equations and matrix-vector multiplications can be translated as the ground states of the constructed Hamiltonians. Based on the variational quantum algorithms, we introduce Hamiltonian morphing together with an adaptive ansätz for efficiently finding the ground state, and show the solution verification. Our algorithms are especially suitable for linear algebra problems with sparse matrices, and have wide applications in machine learning and optimisation problems. The algorithm for matrix multiplications can be also used for Hamiltonian simulation and open system simulation. We evaluate the cost and effectiveness of our algorithm through numerical simulations for solving linear systems of equations. We implement the algorithm on the IBM quantum cloud device with a high solution fidelity of 99.95%.
A novel class of hybrid quantum-classical algorithms based on the variational approach have recently emerged from separate proposals addressing, for example, quantum chemistry and combinatorial problems. These algorithms provide an approximate solution to the problem at hand by encoding it in the state of a quantum computer. The operations used to prepare the state are not a priori fixed but, quite the opposite, are subjected to a classical optimization procedure that modifies the quantum gates and improves the quality of the approximate solution. While the quantum hardware determines the size of the problem and what states are achievable (limited, respectively, by the number of qubits and by the kind and number of possible quantum gates), it is the classical optimization procedure that determines the way in which the quantum states are explored and whether the best available solution is actually reached. In addition, the quantities required in the optimization, for example the objective function itself, have to be estimated with finite precision in any experimental implementation. While it is desirable to have very precise estimates, this comes at the cost of repeating the state preparation and measurement multiple times. Here we analyze the competing requirements of high precision and low number of repetitions and study how the overall performance of the variational algorithm is affected by the precision level and the choice of the optimization method. Finally, this study introduces quasi-Newton optimization methods in the general context of hybrid variational algorithms and presents quantitative results for the Quantum Approximate Optimization Algorithm.
Solving linear systems of equations is central to many engineering and
scientific fields. Several quantum algorithms have been proposed for linear
systems, where the goal is to prepare ∣x⟩ such that A∣x⟩∝∣b⟩. While these algorithms are promising, the time horizon for their
implementation is long due to the required quantum circuit depth. In this work,
we propose a variational hybrid quantum-classical algorithm for solving linear
systems, with the aim of reducing the circuit depth and doing much of the
computation classically. We propose a cost function based on the overlap
between ∣b⟩ and A∣x⟩, and we derive an operational meaning for
this cost in terms of the solution precision ϵ. We also introduce a
quantum circuit to estimate this cost, while showing that this cost cannot be
efficiently estimated classically. Using Rigetti's quantum computer, we
successfully implement our algorithm up to a problem size of 32×32.
Furthermore, we numerically find that the complexity of our algorithm scales
efficiently in both 1/ϵ and κ, with κ the condition
number of A. Our algorithm provides a heuristic for quantum linear systems
that could make this application more near term.