We introduce a novel hybrid algorithm to simulate the real-time evolution of quantum systems using parameterized quantum circuits. The method, named "projected - Variational Quantum Dynamics" (p-VQD) realizes an iterative, global projection of the exact time evolution onto the parameterized manifold. In the small time-step limit, this is equivalent to the McLachlan's variational principle. Our approach is efficient in the sense that it exhibits an optimal linear scaling with the total number of variational parameters. Furthermore, it is global in the sense that it uses the variational principle to optimize all parameters at once. The global nature of our approach then significantly extends the scope of existing efficient variational methods, that instead typically rely on the iterative optimization of a restricted subset of variational parameters. Through numerical experiments, we also show that our approach is particularly advantageous over existing global optimization algorithms based on the time-dependent variational principle that, due to a demanding quadratic scaling with parameter numbers, are unsuitable for large parameterized quantum circuits.
Mitarai, Negoro, Kitagawa, and Fujii proposed a type of parameterized quantum circuits, for which they gave a way to estimate derivatives wrt the parameters using only changes in the values of the parameters, not in the circuit itself, i.e., no ancillas or controlled operations. Recently, Schuld et al. have extended the results, but they need to revert to ancillas and controlled operations for some cases. In this note, we extend the types of MiNKiF circuits for which derivatives can be computed without ancillas or controlled operations --- at the cost of a larger number of evaluation points. W…
Parameterized quantum circuits play an essential role in the performance of many variational hybrid quantum-classical (HQC) algorithms. One challenge in implementing such algorithms is to choose an effective circuit that well represents the solution space while maintaining a low circuit depth and number of parameters. To characterize and identify expressible, yet compact, parameterized circuits, we propose several descriptors, including measures of expressibility and entangling capability, that can be statistically estimated from classical simulations of parameterized quantum circuits. We comp…
Parameterized circuit instantiation is a common technique encountered in the generation of circuits for a large class of hybrid quantum-classical algorithms. Despite being supported by popular quantum compilation infrastructures such as IBM Qiskit and Google Cirq, instantiation has not been extensively considered in the context of circuit compilation and optimization pipelines. In this work, we describe algorithms to apply instantiation during two common compilation steps: circuit optimization and gate-set transpilation. When placed in a compilation workflow, our circuit optimization algorithm…
Noisy intermediate scale quantum computers are useful for various tasks such as state preparation and variational quantum algorithms. However, the non-Euclidean quantum geometry of parameterized quantum circuits is detrimental for these applications. Here, we introduce the natural parameterized quantum circuit (NPQC) that can be initialised with a Euclidean quantum geometry. The initial training of variational quantum algorithms is substantially sped up as the gradient is equivalent to the quantum natural gradient. Further, we show how to estimate the parameters of the NPQC by sampling the cir…
Parameterized quantum circuits (PQC, aka, variational quantum circuits) are among the proposals for a computational advantage over classical computation of near-term (not fault tolerant) digital quantum computers. PQCs have to be "trained" -- i.e., the expectation value function has to be maximized over the space of parameters. This paper deals with the number of samples (or "runs" of the quantum computer) which are required to train the PQC, and approaches it from an information theoretic viewpoint. The main take-away is a disparity in the large amount of information contained in a single exa…
The topic area of this paper parameterized quantum circuits (quantum neural networks) which are trained to estimate a given function, specifically the type of circuits proposed by Mitarai et al. (Phys. Rev. A, 2018). The input is encoded into amplitudes of states of qubits. The no-cloning principle of quantum mechanics suggests that there is an advantage in redundantly encoding the input value several times. We follow this suggestion and prove lower bounds on the number of redundant copies for two types of input encoding. We draw conclusions for the architecture design of QNNs.
Variational quantum algorithms have been introduced as a promising class of quantum-classical hybrid algorithms that can already be used with the noisy quantum computing hardware available today by employing parameterized quantum circuits. Considering the non-trivial nature of quantum circuit compilation and the subtleties of quantum computing, it is essential to verify that these parameterized circuits have been compiled correctly. Established equivalence checking procedures that handle parameter-free circuits already exist. However, no methodology capable of handling circuits with parameters…
Taylor University（泰勒大学）；Oak Ridge National Laboratory（橡树岭国家实验室）
Variational training of parameterized quantum circuits (PQCs) underpins many workflows employed on near-term noisy intermediate scale quantum (NISQ) devices. It is a hybrid quantum-classical approach that minimizes an associated cost function in order to train a parameterized ansatz. In this paper we adapt the qualitative loss landscape characterization for neural networks introduced in \cite{goodfellow2014qualitatively,li2017visualizing} and tests for connectivity used in \cite{draxler2018essentially} to study the loss landscape features in PQC training. We present results for PQCs trained on…
Scaling of variational quantum algorithms to large problem sizes requires efficient optimization of random parameterized quantum circuits. For such circuits with uncorrelated parameters, the presence of exponentially vanishing gradients in cost function landscapes is an obstacle to optimization by gradient descent methods. In this work, we prove that reducing the dimensionality of the parameter space by utilizing circuit modules containing spatially or temporally correlated gate layers can allow one to circumvent the vanishing gradient phenomenon. Examples are drawn from random separable circu…
We propose an efficient method for simultaneously optimizing both the structure and parameter values of quantum circuits with only a small computational overhead. Shallow circuits that use structure optimization perform significantly better than circuits that use parameter updates alone, making this method particularly suitable for noisy intermediate-scale quantum computers. We demonstrate the method for optimizing a variational quantum eigensolver for finding the ground states of Lithium Hydride and the Heisenberg model in simulation, and for finding the ground state of Hydrogen gas on the IB…
An active area of investigation in the search for quantum advantage is Quantum Machine Learning. Quantum Machine Learning, and Parameterized Quantum Circuits in a hybrid quantum-classical setup in particular, could bring advancements in accuracy by utilizing the high dimensionality of the Hilbert space as feature space. But is the ability of a quantum circuit to uniformly address the Hilbert space a good indicator of classification accuracy? In our work, we use methods and quantifications from prior art to perform a numerical study in order to evaluate the level of correlation. We find a stron…
Cambridge Quantum Computing Limited（剑桥量子计算有限公司）；University College London（伦敦大学学院）
Hybrid quantum-classical systems make it possible to utilize existing quantum computers to their fullest extent. Within this framework, parameterized quantum circuits can be regarded as machine learning models with remarkable expressive power. This Review presents the components of these models and discusses their application to a variety of data-driven tasks, such as supervised learning and generative modeling. With an increasing number of experimental demonstrations carried out on actual quantum hardware and with software being actively developed, this rapidly growing field is poised to have…
Parameterized quantum circuits (PQCs) are a central component of many variational quantum algorithms, yet there is a lack of understanding of how their parameterization impacts algorithm performance. We initiate this discussion by using principal bundles to geometrically characterize two-qubit PQCs. On the base manifold, we use the Mannoury-Fubini-Study metric to find a simple equation relating the Ricci scalar (geometry) and concurrence (entanglement). By calculating the Ricci scalar during a variational quantum eigensolver (VQE) optimization process, this offers us a new perspective to how a…
Machine learning is seen as a promising application of quantum computation. For near-term noisy intermediate-scale quantum (NISQ) devices, parametrized quantum circuits (PQCs) have been proposed as machine learning models due to their robustness and ease of implementation. However, the cost function is normally calculated classically from repeated measurement outcomes, such that it is no longer encoded in a quantum state. This prevents the value from being directly manipulated by a quantum computer. To solve this problem, we give a routine to embed the cost function for machine learning into a…
In recent years, parameterized quantum circuits have become a major tool to design quantum algorithms for optimization problems. The challenge in fully taking advantage of a given family of parameterized circuits lies in finding a good set of parameters in a non-convex landscape that can grow exponentially to the number of parameters. We introduce a new framework for optimizing parameterized quantum circuits: round SDP solutions to circuit parameters. Within this framework, we propose an algorithm that produces approximate solutions for a quantum optimization problem called Quantum Max Cut. Th…
Current structure optimization algorithms optimize the structure of quantum circuit from scratch for each new task of variational quantum algorithms (VQAs) without using any prior experience, which is inefficient and time-consuming. Besides, the number of quantum gates is a hyper-parameter of these algorithms, which is difficult and time-consuming to determine. In this paper, we propose a rapid structure optimization algorithm for VQAs which automatically determines the number of quantum gates and directly generates the optimal structures for new tasks with the meta-trained graph variational a…
UBTECH Sydney Artificial Intelligence Centre（优必选悉尼人工智能中心）；The University of Sydney（悉尼大学）；University of Technology Sydney（悉尼科技大学）
Parameterized quantum circuits (PQCs) have been broadly used as a hybrid quantum-classical machine learning scheme to accomplish generative tasks. However, whether PQCs have better expressive power than classical generative neural networks, such as restricted or deep Boltzmann machines, remains an open issue. In this paper, we prove that PQCs with a simple structure already outperform any classical neural network for generative tasks, unless the polynomial hierarchy collapses. Our proof builds on known results from tensor networks and quantum circuits (in particular, instantaneous quantum poly…
Variational Quantum Algorithms (VQAs) have emerged as a powerful class of algorithms that is highly suitable for noisy quantum devices. Therefore, investigating their design has become key in quantum computing research. Previous works have shown that choosing an effective parameterized quantum circuit (PQC) or ansatz for VQAs is crucial to their overall performance, especially on near-term devices. In this paper, we utilize pulse-level access to quantum machines and our understanding of their two-qubit interactions to optimize the design of two-qubit entanglers in a manner suitable for VQAs. O…
Quantum machine learning (QML) is the use of quantum computing for the computation of machine learning algorithms. With the prevalence and importance of classical data, a hybrid quantum-classical approach to QML is called for. Parameterized Quantum Circuits (PQCs), and particularly Quantum Kernel PQCs, are generally used in the hybrid approach to QML. In this paper we discuss some important aspects of PQCs with quantum kernels including PQCs, quantum kernels, quantum kernels with quantum advantage, and the trainability of quantum kernels. We conclude that quantum kernels with hybrid kernel met…
The variational quantum eigensolver is a prominent hybrid quantum-classical algorithm expected to impact near-term quantum devices. They are usually based on a circuit ansatz consisting of parameterized single-qubit gates and fixed two-qubit gates. We study the effect of parameterized two-qubit gates in the variational quantum eigensolver. We simulate a variational quantum eigensolver algorithm using fixed and parameterized two-qubit gates in the circuit ansatz and show that the parameterized versions outperform the fixed versions, both when it comes to best energy and reducing outliers, for a…
The University of Tokyo（东京大学）；Austrian Academy of Sciences（奥地利科学院）；Technische Universität Wien（维也纳工业大学）
Optimizing parameterized quantum circuits is a key routine in using near-term quantum devices. However, the existing algorithms for such optimization require an excessive number of quantum-measurement shots for estimating expectation values of observables and repeating many iterations, whose cost has been a critical obstacle for practical use. We develop an efficient alternative optimization algorithm, stochastic gradient line Bayesian optimization (SGLBO), to address this problem. SGLBO reduces the measurement-shot cost by estimating an appropriate direction of updating circuit parameters bas…
