It is NP-complete to find non-negative factors W and H with fixed rank r from a non-negative matrix X by minimizing ||X-WH^Τ ||^2. Although the separability assumption (all data points are in the conical hull of the extreme rows) enables polynomial-time algorithms, the computational cost is not affordable for big data. This paper investigates how the power of quantum computation can be capitalized to solve the non-negative matrix factorization with the separability assumption (SNMF) by devising a quantum algorithm based on the divide-and-conquer anchoring (DCA) scheme [Zhou et al., 2013]. The design of quantum DCA (QDCA) is challenging. In the divide step, the random projections in DCA is completed by a quantum algorithm for linear operations, which achieves the exponential speedup. We then devise a heuristic post-selection procedure which extracts the information of anchors stored in the quantum states efficiently. Under a plausible assumption, QDCA performs efficiently, achieves the quantum speedup, and is beneficial for high dimensional problems.
It is NP-complete to find non-negative factors W and H with fixed rank r from a non-negative matrix X by minimizing ||X-WH^Τ ||^2. Although the separability assumption (all data points are in the conical hull of the extreme rows) enables polynomial-time algorithms, the computational cost is not affordable for big data. This paper investigates how the power of quantum computation can be capitalized to solve the non-negative matrix factorization with the separability assumption (SNMF) by devising a quantum algorithm based on the divide-and-conquer anchoring (DCA) scheme [Zhou et al., 2013]. The design of quantum DCA (QDCA) is challenging. In the divide step, the random projections in DCA is completed by a quantum algorithm for linear operations, which achieves the exponential speedup. We then devise a heuristic post-selection procedure which extracts the information of anchors stored in the quantum states efficiently. Under a plausible assumption, QDCA performs efficiently, achieves the quantum speedup, and is beneficial for high dimensional problems.
Among matrix decomposition methods, the two factorized matrices obtained by Non-negative matrix factorization (NMF) generally yield a more natural and interpretable representations. It is computationally expensive to find the two factorized matrices W and H with a fixed rank r from a non-negative matrix X∈Rn⇥m by minimizing ||X-WH^Τ ||^2. Even if the proposed separability assumption enables polynomial separable NMF (SNMF) algorithms, the computational cost is not affordable for the continuously growing big data. As a powerful technique to deal with high dimensional data, quantum computing possesses quantum advantages for solving machine learning problems, at which the quantum advantages include both an exponential speedup and a more efficient learning ability with respect to the classical counterparts. In this research, based on the divide-and-conquer anchoring (DCA) scheme, quantum DCA (QDCA) is devised to solve SNMF problems. Benefiting from the linear algebraic nature of quantum mechanics and the convenience that statistical distribution is natively available in quantum mechanics with a superset, QDCA achieves an exponential speedup with respect to the classical counterpart. The runtime complexity is O(poly log(n + m)), where n x m is the size of the input matrix. In addition, the proposed QDCA can naturally generalize to solve near-separable NMF under a polynomial logarithmic runtime. Specifically, the time consuming divide step can be completed by a quantum algorithm for linear operations with an exponential speedup. In the conquer step, considering that reading out quantum data into classical forms is exponential expensive and attributing to that anchors can be sampled with a high probability from a probability distribution in the resulting quantum states, the proposed heuristic post-selection method enables QDCA to extract information from quantum states efficiently, which also maintains the quantum speedup after measurements. Under a plausible assumption, QDCA performs efficiently, achieves the exponential speedup, and is beneficial for high dimensional problems. The QDCA is a hybrid quantum and classical algorithm that maintains the quantum speedup after measurements, which provides a new way to develop quantum machine learning algorithms.
Non-negative Matrix Factorization (NMF) asks to decompose a (entry-wise) non-negative matrix into the product of two smaller-sized nonnegative matrices, which has been shown intractable in general. In order to overcome this issue, separability assumption is introduced which assumes all data points are in a conical hull. This assumption makes NMF tractable and widely used in text analysis and image processing, but still impractical for huge-scale datasets. In this paper, inspired by recent development on dequantizing techniques, we propose a new classical algorithm for separable NMF problem. Our new algorithm runs in polynomial time in the rank and logarithmic in the size of input matrices, which achieves an exponential speedup in the low-rank setting.
Abstract Noisy linear problems have been studied in various science and engineering disciplines. A class of ‘hard’ noisy linear problems can be formulated as follows: Given a matrix A ^ and a vector b constructed using a finite set of samples, a hidden vector or structure involved in b is obtained by solving a noise-corrupted linear equation A ^ x ≈ b + η , where η is a noise vector that cannot be identified. For solving such a noisy linear problem, we consider a quantum algorithm based on a divide-and-conquer strategy, wherein a large core process is divided into smaller subprocesses. The algorithm appropriately reduces both the computational complexities and size of a quantum sample. More specifically, if a quantum computer can access a particular reduced form of the quantum samples, polynomial quantum-sample and time complexities are achieved in the main computation. The size of a quantum sample and its executing system can be reduced, e.g., from exponential to sub-exponential with respect to the problem length, which is better than other results we are aware. We analyse the noise model conditions for such a quantum advantage, and show when the divide-and-conquer strategy can be beneficial for quantum noisy linear problems.
Quantum optimization as a field has largely been restricted by the constraints of current quantum computing hardware, as limitations on size, performance, and fidelity mean most non-trivial problem instances won't fit on quantum devices. Even proposed solutions such as distributed quantum computing systems may struggle to achieve scale due to the high cost of inter-device communication. To address these concerns, we propose Deferred Constraint Quantum Divide and Conquer Algorithm (DC-QDCA), a method for constructing quantum circuits which greatly reduces inter-device communication costs for some quantum graph optimization algorithms. This is achieved by identifying a set of vertices whose removal partitions the input graph, known as a separator; by manipulating the placement of constraints associated with the vertices in the separator, we can greatly simplify the topology of the optimization circuit, reducing the number of required inter-device operations. Furthermore, we introduce an iterative algorithm which builds on these techniques to find solutions for problems with potentially thousands of variables. Our experimental results using quantum simulators have shown that we can construct tractable circuits nearly three times the size of previous QDCA methods while retaining a similar or greater level of quality.
Advantages in several fields of research and industry are expected with the rise of quantum computers. However, the computational cost to load classical data in quantum computers can impose restrictions on possible quantum speedups. Known algorithms to create arbitrary quantum states require quantum circuits with depth O(N) to load an N-dimensional vector. Here, we show that it is possible to load an N-dimensional vector with exponential time advantage using a quantum circuit with polylogarithmic depth and entangled information in ancillary qubits. Results show that we can efficiently load data in quantum devices using a divide-and-conquer strategy to exchange computational time for space. We demonstrate a proof of concept on a real quantum device and present two applications for quantum machine learning. We expect that this new loading strategy allows the quantum speedup of tasks that require to load a significant volume of information to quantum devices.
Quantum optimization as a field has largely been restricted by the constraints of current quantum computing hardware, as limitations on size, performance, and fidelity mean most non-trivial problem instances won't fit on quantum devices. Even proposed solutions such as distributed quantum computing systems may struggle to achieve scale due to the high cost of inter-device communication. To address these concerns, we propose Deferred Constraint Quantum Divide and Conquer Algorithm (DC-QDCA), a method for constructing quantum circuits which greatly reduces inter-device communication costs for some quantum graph optimization algorithms. This is achieved by identifying a set of vertices whose removal partitions the input graph, known as a separator; by manipulating the placement of constraints associated with the vertices in the separator, we can greatly simplify the topology of the optimization circuit, reducing the number of required inter-device operations. Furthermore, we introduce an iterative algorithm which builds on these techniques to find solutions for problems with potentially thousands of variables. Our experimental results using quantum simulators have shown that we can construct tractable circuits nearly three times the size of previous QDCA methods while retaining a similar or greater level of quality.