Abstract The trace norm of matrices plays an important role in quantum information and quantum computing. How to quantify it in today’s noisy intermediate scale quantum (NISQ) devices is a crucial task for information processing. In this paper, we present three variational quantum algorithms on NISQ devices to estimate the trace norms corresponding to different situations. Compared with the previous methods, our means greatly reduce the requirement for quantum resources. Numerical experiments are provided to illustrate the effectiveness of our algorithms.
Abstract The trace norm of matrices plays an important role in quantum information and quantum computing. How to quantify it in today’s noisy intermediate scale quantum (NISQ) devices is a crucial task for information processing. In this paper, we present three variational quantum algorithms on NISQ devices to estimate the trace norms corresponding to different situations. Compared with the previous methods, our means greatly reduce the requirement for quantum resources. Numerical experiments are provided to illustrate the effectiveness of our algorithms.
Variational quantum algorithms (VQAs) are the leading candidates for practical applications of noisy intermediate-scale quantum (NISQ) devices. Recently, pulse-based variational quantum algorithms (p-VQAs) have emerged as a compelling alternative to traditional gate-based VQAs. By optimizing continuous control fields directly on hardware-native Hamiltonians, pulse-based methods are easy to implement and can significantly reduce quantum resource overhead, thereby enhancing the overall performance of quantum algorithms on NISQ devices. In this review, we present a comprehensive survey over recent advances in p-VQAs, covering theory, algorithms, and hardware deployments across various quantum computing platforms. We also review applications of p-VQAs to variational quantum eigensolver in quantum chemistry, quantum approximate optimization algorithm for combinatorial optimization, and quantum machine learning tasks.
Fighting against noise is crucial for Noisy Intermediate Scale Quantum (NISQ) devices to demonstrate practical quantum applications. In this work, we give a paradigm of quantum error mitigation based on the vectorization of density matrices for variational quantum algorithms. Different from the ideas of existing quantum error mitigation methods that try to distill noiseless information from noisy quantum states, our proposal directly changes the way of encoding information and maps the density matrices of noisy quantum states to noiseless virtual pure states, which is realized by an NISQ-friendly measurement protocol and a classical postprocessing procedure. Our protocol requires no knowledge of the noise model, no ability to tune the noise strength, and no ancilla qubits for complicated controlled unitaries. Under our encoding, NISQ devices are always preparing virtual pure quantum states that are highly desired resources for variational quantum algorithms to have good performance in many tasks. We show how this protocol can be well fitted into variational quantum algorithms. We give several concrete ansatz constructions that are suitable for our proposal and do theoretical analysis on the sampling complexity, the expressibility, and the trainability. We also give a discussion on how this protocol is influenced by large noise and how it can be well combined with other quantum error mitigation protocols. The effectiveness of our proposal is demonstrated by various numerical experiments.
Abstract \noindent This paper applies the Variational Quantum Linear Solver (VQLS) to the continuous-time Lyapunov equation ATX+XA=−Q. Using the Kronecker product formulation, the matrix equation is transformed into the linear system Hz=b, where H=In⊗AT+AT⊗In and z=vec(X). VQLS approximates the solution via parameterized quantum circuits optimized in a hybrid quantum-classical loop. Numerical experiments on four control system benchmarks with Kronecker dimensions up to 25×25 demonstrate residual norms as low as 10−9, with logarithmic qubit scaling (2-5 qubits) and shallow circuit depths (6-14 layers) suitable for near-term quantum devices. Compared to the HHL algorithm, VQLS offers reduced circuit complexity and qubit overhead, making it practical for noisy intermediate-scale quantum (NISQ) hardware. This paper demonstrates the potential of variational quantum algorithms for solving Lyapunov equations in control and dynamical systems applications.
Abstract Quantum entanglement is a key resource in quantum technology, and its quantification is a vital task in the current noisy intermediate-scale quantum (NISQ) era. This paper combines hybrid quantum-classical computation and quasi-probability decomposition to propose two variational quantum algorithms, called variational entanglement detection (VED) and variational logarithmic negativity estimation (VLNE), for detecting and quantifying entanglement on near-term quantum devices, respectively. VED makes use of the positive map criterion and works as follows. Firstly, it decomposes a positive map into a combination of quantum operations implementable on near-term quantum devices. It then variationally estimates the minimal eigenvalue of the final state, obtained by executing these implementable operations on the target state and averaging the output states. Deterministic and probabilistic methods are proposed to compute the average. At last, it asserts that the target state is entangled if the optimized minimal eigenvalue is negative. VLNE builds upon a linear decomposition of the transpose map into Pauli terms and the recently proposed trace distance estimation algorithm. It variationally estimates the well-known logarithmic negativity entanglement measure and could be applied to quantify entanglement on near-term quantum devices. Experimental and numerical results on the Bell state, isotropic states, and Breuer states show the validity of the proposed entanglement detection and quantification methods.
Abstract Estimating the difference between quantum data is crucial in quantum computing. However, as typical characterizations of quantum data similarity, the trace distance and quantum fidelity are believed to be exponentially-hard to evaluate in general. In this work, we introduce hybrid quantum–classical algorithms for these two distance measures on near-term quantum devices where no assumption of input state is required. First, we introduce the variational trace distance estimation (VTDE) algorithm. We in particular provide the technique to extract the desired spectrum information of any Hermitian matrix by local measurement. A novel variational algorithm for trace distance estimation is then derived from this technique, with the assistance of a single ancillary qubit. Notably, VTDE could avoid the barren plateau issue with logarithmic depth circuits due to a local cost function. Second, we introduce the variational fidelity estimation algorithm. We combine Uhlmann’s theorem and the freedom in purification to translate the estimation task into an optimization problem over a unitary on an ancillary system with fixed purified inputs. We then provide a purification subroutine to complete the translation. Both algorithms are verified by numerical simulations and experimental implementations, exhibiting high accuracy for randomly generated mixed states.
Parameterized quantum circuits (PQCs) play an essential role in the application of variational quantum algorithms (VQAs) in noisy intermediate-scale quantum (NISQ) devices. The PQCs are a leading candidate to achieve a quantum advantage in NISQ devices and have already been applied in various domains such as quantum chemistry, quantum machine learning, combinatorial optimization, and many others. There is no single definitive way to optimize PQCs. The most commonly used methods are based on computing the gradient via the parameter-shift rule to use classical gradient descent (GD) optimizers like Adam, stochastic GD, and others. In addition, sequential single-qubit optimizers have been proposed, such as Rotosolve, Free-Axis Selection (Fraxis), Free-Quaternion Selection (FQS), and hybrid algorithms from the aforementioned optimizers. We further develop hybrid algorithms than those represented in the previous work by drawing inspiration from the early stopping method used in classical machine learning. The switch between the optimizers depends on the previous cost function values compared to the previous ones. We introduce two new hybrid algorithms that are more robust and scalable, and they outperform previous hybrid methods in terms of convergence towards the global minima across various cost functions. In addition, we find that they are feasible for NISQ devices with different noise profiles.