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Multi-view data can be generated from diverse sources, by different technologies, and in multiple modalities. In various fields, integrating information from multi-view data has pushed the frontier of discovery. In this paper, we develop a new approach for multi-view clustering, which overcomes the limitations of existing methods such as the need of pooling data across views, restrictions on the clustering algorithms allowed within each view, and the disregard for complementary information between views. Our new method, called CPS-merge analysis , merges clusters formed by the Cartesian product of single-view cluster labels, guided by the principle of maximizing clustering stability as evaluated by CPS analysis. In addition, we introduce measures to quantify the contribution of each view to the formation of any cluster. CPS-merge analysis can be easily incorporated into an existing clustering pipeline because it only requires single-view cluster labels instead of the original data. We can thus readily apply advanced single-view clustering algorithms. Importantly, our approach accounts for both consensus and complementary effects between different views, whereas existing ensemble methods focus on finding a consensus for multiple clustering results, implying that results from different views are variations of one clustering structure. Through experiments on single-cell datasets, we demonstrate that our approach frequently outperforms other state-of-the-art methods. 

The time-marching strategy, which propagates the solution from one time step to the next, is a natural strategy for solving time-dependent differential equations on classical computers, as well as for solving the Hamiltonian simulation problem on quantum computers. For more general homogeneous linear differential equations d d t | ψ ( t ) ⟩ = A ( t ) | ψ ( t ) ⟩ , | ψ ( 0 ) ⟩ = | ψ 0 ⟩ , a time-marching based quantum solver can suffer from exponentially vanishing success probability with respect to the number of time steps and is thus considered impractical. We solve this problem by repeatedly invoking a technique called the uniform singular value amplification, and the overall success probability can be lower bounded by a quantity that is independent of the number of time steps. The success probability can be further improved using a compression gadget lemma. This provides a path of designing quantum differential equation solvers that is alternative to those based on quantum linear systems algorithms (QLSA). We demonstrate the performance of the time-marching strategy with a high-order integrator based on the truncated Dyson series. The complexity of the algorithm depends linearly on the amplification ratio, which quantifies the deviation from a unitary dynamics. We prove that the linear dependence on the amplification ratio attains the query complexity lower bound and thus cannot be improved in the worst case. This algorithm also surpasses existing QLSA based solvers in three aspects: (1) A ( t ) does not need to be diagonalizable. (2) A ( t ) can be non-smooth, and is only of bounded variation. (3) It can use fewer queries to the initial state | ψ 0 ⟩ . Finally, we demonstrate the time-marching strategy with a first-order truncated Magnus series, which simplifies the implementation compared to high-order truncated Dyson series approach, while retaining the aforementioned benefits. Our analysis also raises some open questions concerning the differences between time-marching and QLSA based methods for solving differential equations. 

Abstract Hamiltonian simulation is one of the most important problems in quantum computation, and quantum singular value transformation (QSVT) is an efficient way to simulate a general class of Hamiltonians. However, the QSVT circuit typically involves multiple ancilla qubits and multi-qubit control gates. In order to simulate a certain class of n -qubit random Hamiltonians, we propose a drastically simplified quantum circuit that we refer to as the minimal QSVT circuit, which uses only one ancilla qubit and no multi-qubit controlled gates. We formulate a simple metric called the quantum unitary evolution score (QUES), which is a scalable quantum benchmark and can be verified without any need for classical computation. Under the globally depolarized noise model, we demonstrate that QUES is directly related to the circuit fidelity, and the potential classical hardness of an associated quantum circuit sampling problem. Under the same assumption, theoretical analysis suggests there exists an ‘optimal’ simulation time t opt  ≈ 4.81, at which even a noisy quantum device may be sufficient to demonstrate the potential classical hardness. 

Symmetric quantum signal processing provides a parameterized representation of a real polynomial, which can be translated into an efficient quantum circuit for performing a wide range of computational tasks on quantum computers. For a given polynomial f , the parameters (called phase factors) can be obtained by solving an optimization problem. However, the cost function is non-convex, and has a very complex energy landscape with numerous global and local minima. It is therefore surprising that the solution can be robustly obtained in practice, starting from a fixed initial guess Φ 0 that contains no information of the input polynomial. To investigate this phenomenon, we first explicitly characterize all the global minima of the cost function. We then prove that one particular global minimum (called the maximal solution) belongs to a neighborhood of Φ 0 , on which the cost function is strongly convex under the condition ‖ f ‖ ∞ = O ( d − 1 ) with d = d e g ( f ) . Our result provides a partial explanation of the aforementioned success of optimization algorithms.