Parameterized Quantum Circuits Research Articles

Quantum phase recognition using quantum tensor networks

Machine learning (ML) has recently facilitated many advances in solving problems related to many-body physical systems. Given the intrinsic quantum nature of these problems, it is natural to speculate that quantum-enhanced machine learning will enable us to unveil even greater details than we currently have. With this motivation, this paper examines a quantum machine learning approach based on shallow variational ansatz inspired by tensor networks for supervised learning tasks. In particular, we first look at the standard image classification tasks using the Fashion-MNIST dataset and study the effect of repeating tensor network layers on ansatz’s expressibility and performance. Finally, we use this strategy to tackle the problem of quantum phase recognition for the transverse-field Ising and Heisenberg spin models in one and two dimensions, where we were able to reach \(\ge 98\%\) test-set accuracies with both multi-scale entanglement renormalization ansatz (MERA) and tree tensor network (TTN) inspired parametrized quantum circuits.

Avoiding barren plateaus via transferability of smooth solutions in a Hamiltonian variational ansatz

Parametrized quantum circuits inspired by adiabatic quantum computation often suffer from vanishing gradients, hindering the trainability of hybrid quantum-classical algorithms. Here, the authors put forward a strategy to overcome this by reusing parameters and iterating from small to large systems.

Variational Quantum Simulation of Valence-Bond Solids

We introduce a hybrid quantum-classical variational algorithm to simulate ground-state phase diagrams of frustrated quantum spin models in the thermodynamic limit. The method is based on a cluster-Gutzwiller ansatz where the wave function of the cluster is provided by a parameterized quantum circuit whose key ingredient is a two-qubit real XY gate allowing to efficiently generate valence-bonds on nearest-neighbor qubits. Additional tunable single-qubit Z- and two-qubit ZZ-rotation gates allow the description of magnetically ordered and paramagnetic phases while restricting the variational optimization to the U(1) subspace. We benchmark the method against the J1−J2 Heisenberg model on the square lattice and uncover its phase diagram, which hosts long-range ordered Neel and columnar anti-ferromagnetic phases, as well as an intermediate valence-bond solid phase characterized by a periodic pattern of 2×2 strongly-correlated plaquettes. Our results show that the convergence of the algorithm is guided by the onset of long-range order, opening a promising route to synthetically realize frustrated quantum magnets and their quantum phase transition to paramagnetic valence-bond solids with currently developed superconducting circuit devices.

Hamiltonian-based data loading with shallow quantum circuits

Data loading with shallow quantum circuits is a highly desirable ingredient for efficient execution of many quantum algorithms before large-scale quantum error corrections with full fault tolerance become readily available. The need for efficient data loading is especially urgent for the study of quantum machine learning. In this work, we propose a protocol that only uses a parameterized shallow quantum circuit without ancilla qubits for loading data into the amplitude of a quantum state with high fidelity. We term this data-loading method Hamiltonian-based data loading (HDL), which comprises two stages. First, the HDL algorithm identifies a Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$ whose unique ground state $|\ensuremath{\psi}\ensuremath{\rangle}$ represents the normalized data $\stackrel{P\vec}{x}$ in the form of amplitude encoding. Next, the target state is reconstructed with a parameterized quantum circuit by utilizing methods like variational quantum eigensolver to minimize energy. In this work, we provide three convincing examples to demonstrate the effectiveness of HDL for loading an $N$-dimensional classical data with minimal quantum resources ($O[\mathrm{poly}({log}_{2}N)]$-depth quantum circuit without ancilla qubits). Our approach is particularly useful for quantum hardware without large-scale error corrections, and shall benefit the development of quantum machine learning algorithms.

Circuit depth scaling for quantum approximate optimization

Variational quantum algorithms are the centerpiece of modern quantum programming. These algorithms involve training parameterized quantum circuits using a classical co-processor, an approach adapted partly from classical machine learning. An important subclass of these algorithms, designed for combinatorial optimization on currrent quantum hardware, is the quantum approximate optimization algorithm (QAOA). It is known that problem density - a problem constraint to variable ratio - induces under-parametrization in fixed depth QAOA. Density dependent performance has been reported in the literature, yet the circuit depth required to achieve fixed performance (henceforth called critical depth) remained unknown. Here, we propose a predictive model, based on a logistic saturation conjecture for critical depth scaling with respect to density. Focusing on random instances of MAX-2-SAT, we test our predictive model against simulated data with up to 15 qubits. We report the average critical depth, required to attain a success probability of 0.7, saturates at a value of 10 for densities beyond 4. We observe the predictive model to describe the simulated data within a $3\sigma$ confidence interval. Furthermore, based on the model, a linear trend for the critical depth with respect problem size is recovered for the range of 5 to 15 qubits.

Fast-forwarding quantum simulation with real-time quantum Krylov subspace algorithms

Quantum subspace diagonalization (QSD) algorithms have emerged as a competitive family of algorithms that avoid many of the optimization pitfalls associated with parameterized quantum circuit algorithms. While the vast majority of the QSD algorithms have focused on solving the eigenpair problem for ground, excited-state, and thermal observable estimation, there has been a lot less work in considering QSD algorithms for the problem of quantum dynamical simulation. In this work, we propose several quantum Krylov fast-forwarding (QKFF) algorithms capable of predicting long-time dynamics well beyond the coherence time of current quantum hardware. Our algorithms use real-time evolved Krylov basis states prepared on the quantum computer and a multi-reference subspace method to ensure convergence towards high-fidelity, long-time dynamics. In particular, we show that the proposed multi-reference methodology provides a systematic way of trading off circuit depth with classical post-processing complexity. We also demonstrate the efficacy of our approach through numerical implementations for several quantum chemistry problems including the calculation of the auto-correlation and dipole moment correlation functions

Hamiltonian reconstruction as metric for variational studies

Variational approaches are among the most powerful techniques to approximately solve quantum many-body problems. These encompass both variational states based on tensor or neural networks, and parameterized quantum circuits in variational quantum eigensolvers. However, self-consistent evaluation of the quality of variational wavefunctions is a notoriously hard task. Using a recently developed Hamiltonian reconstruction method, we propose a multi-faceted approach to evaluating the quality of neural-network based wavefunctions. Specifically, we consider convolutional neural network (CNN) and restricted Boltzmann machine (RBM) states trained on a square lattice spin-1/21/2J_1\!-\!J_2J1−J2 Heisenberg model. We find that the reconstructed Hamiltonians are typically less frustrated, and have easy-axis anisotropy near the high frustration point. In addition, the reconstructed Hamiltonians suppress quantum fluctuations in the large J_2J2 limit. Our results highlight the critical importance of the wavefunction’s symmetry. Moreover, the multi-faceted insight from the Hamiltonian reconstruction reveals that a variational wave function can fail to capture the true ground state through suppression of quantum fluctuations.

Non-trivial symmetries in quantum landscapes and their resilience to quantum noise

Very little is known about the cost landscape for parametrized Quantum Circuits (PQCs). Nevertheless, PQCs are employed in Quantum Neural Networks and Variational Quantum Algorithms, which may allow for near-term quantum advantage. Such applications require good optimizers to train PQCs. Recent works have focused on quantum-aware optimizers specifically tailored for PQCs. However, ignorance of the cost landscape could hinder progress towards such optimizers. In this work, we analytically prove two results for PQCs: (1) We find an exponentially large symmetry in PQCs, yielding an exponentially large degeneracy of the minima in the cost landscape. Alternatively, this can be cast as an exponential reduction in the volume of relevant hyperparameter space. (2) We study the resilience of the symmetries under noise, and show that while it is conserved under unital noise, non-unital channels can break these symmetries and lift the degeneracy of minima, leading to multiple new local minima. Based on these results, we introduce an optimization method called Symmetry-based Minima Hopping (SYMH), which exploits the underlying symmetries in PQCs. Our numerical simulations show that SYMH improves the overall optimizer performance in the presence of non-unital noise at a level comparable to current hardware. Overall, this work derives large-scale circuit symmetries from local gate transformations, and uses them to construct a noise-aware optimization method.

Connecting geometry and performance of two-qubit parameterized quantum circuits

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 and why Quantum Natural Gradient outperforms the standard gradient descent. We argue that the key to the Quantum Natural Gradient's superior performance is its ability to find regions of high negative curvature early in the optimization process. These regions of high negative curvature appear to be important in accelerating the optimization process.

Equivalence of quantum barren plateaus to cost concentration and narrow gorges

Optimizing parameterized quantum circuits (PQCs) is the leading approach to make use of near-term quantum computers. However, very little is known about the cost function landscape for PQCs, which hinders progress towards quantum-aware optimizers. In this work, we investigate the connection between three different landscape features that have been observed for PQCs: (1) exponentially vanishing gradients (called barren plateaus (BPs)), (2) exponential cost concentration about the mean, and (3) the exponential narrowness of minima (called narrow gorges). We analytically prove that these three phenomena occur together, i.e., when one occurs then so do the other two. A key implication of this result is that one can numerically diagnose BPs via cost differences rather than via the computationally more expensive gradients. More broadly, our work shows that quantum mechanics rules out certain cost landscapes (which otherwise would be mathematically possible), and hence our results could be interesting from a quantum foundations perspective.

An Image Classification Algorithm Based on Hybrid Quantum Classical Convolutional Neural Network

Quantum machine learning is emerging as a strategy to solve real-world problems. As a quantum computing model, parameterized quantum circuits provide an approach for constructing quantum machine learning algorithms, which may either realize computational acceleration or achieve better algorithm performance than classical algorithms. Based on the parameterized quantum circuit, we propose a hybrid quantum-classical convolutional neural network (HQCCNN) model for image classification that comprises both quantum and classical components. The quantum convolutional layer is designed using a parameterized quantum circuit. It is used to perform linear unitary transformation on the quantum state to extract hidden information. In addition, the quantum pooling unit is used to perform pooling operations. After the evolution of the quantum system, we measure the quantum state and input the measurement results into a classical fully connected layer for further processing. We demonstrate its potential by applying HQCCNN to the MNIST dataset. Compared to a convolutional neural network in a similar architecture, the results reveal that HQCCNN has a faster training speed and higher testing set accuracy than a convolutional neural network.

Parameterized Quantum Circuits for Learning Cooperative Quantum Teleportation

AbstractQuantum teleportation is an elemental process in quantum communication and many variants have been widely investigated theoretically and experimentally. Motivated by cooperation in classical communications, cooperative quantum teleportation (CQT) are developed with features of the cooperation referring to allocation of resources and operations among participants. Parameterized quantum circuit (PQC) are employed to learn the CQT protocols on account of different training scenarios with controls of gate parameters among shared entangled states, measurement, and recovery operations. Numerical results show the CQT protocols can be discovered with unit fidelity regardless of ideal or nonideal channels by training the PQCs with hybrid quantum‐classical (HQC) optimization under full cooperations of the participants. Further, the influence of quantum channel noise on the teleportation performance is explored by parameterizing the entangled states with noise and entanglement parameters under adjustable recovery operations. Simulation results indicate the trained PQC can improve the system performance, which implies the potential denoising capability of the HQC algorithms. The suggested CQT protocols satisfy the underlying properties of universality, randomness, locality and cooperation, and discussions indicate the designed quantum circuits can be optimized in view of quantum and circuit costs by appropriate optimization algorithms for achieving higher state fidelity in the future work.

Developing a Model for Predicting Lung Cancer Using Variational Quantum-Classical Algorithm: A Survey

There are real life problems that currently proved hard for classical computers (even the best of supercomputers) to solve-the so called computationally intractable problems. Computers and computing devices are limited, by virtue of the fact that they cannot perform certain complex problems. For example, modern cryptography assumes that it is impossible to factorize a large number as the complexity of solving this problem increases exponentially. Even the best supercomputers of today cannot sufficiently find the prime factors of a number with 700-1,000 digits. With quantum computing, however, Shor’s algorithm proved that in principle, a quantum computer can be used to break the security of conventional cryptosystems. Quantum computing also promises exponential improvements for many optimization problems. Lov Grover algorithm was also designed to search for an element in unstructured database. Effort is currently on to leverage quantum computing and quantum mechanical phenomena to tackle complex machine learning problems. Using machine learning algorithm to solve problems is common place. What is however new is using quantum machine learning algorithm to solve problems such as detecting, classifying and predicting disease such as lung cancer. In this work, we highlight the limitations of classical computers to solve certain problems and then propose a hybrid model called the variational quantum-classical algorithm to predict the possibility of a patient developing lung cancer given a set of features which are spelt out in the dataset. The dataset is initially pre-processed, and made to pass through a Parameterized Quantum Circuit (PQC) and then post-processed. Finally, the output of the post-processing phase is used for prediction of lung cancer.

Quantum Architecture Search with Meta‐Learning

AbstractVariational quantum algorithms (VQAs) have been successfully applied to quantum approximate optimization algorithms, variational quantum compiling and quantum machine learning models. The performances of VQAs largely depend on the architecture of parameterized quantum circuits (PQCs). Quantum architecture search (QAS) aims to automate the design of PQCs in different VQAs with classical optimization algorithms. However, current QAS algorithms do not use prior experiences and search the quantum architecture from scratch for each new task, which is inefficient and time consuming. In this paper, a meta quantum architecture search (MetaQAS) algorithm is proposed, which learns good initialization heuristics of the architecture (i.e., meta‐architecture), along with the meta‐parameters of quantum gates from a number of training tasks such that they can adapt to new tasks with fewer gradient updates, which leads to fast learning on new tasks. The proposed MetaQAS can be used with arbitrary gradient‐based QAS algorithms. Simulation results on variational quantum compiling (VQC) and quantum approximate optimization algorithm (QAOA) show that the architectures optimized by MetaQAS converge faster than a state‐of‐the‐art gradient‐based QAS algorithm, namely DQAS. MetaQAS also achieves a better solution than DQAS after fine‐tuning of gate parameters.

Simulating time evolution with fully optimized single-qubit gates on parametrized quantum circuits

We propose a method to sequentially optimize arbitrary single-qubit gates in parametrized quantum circuits for simulating real- and imaginary-time evolution. The method utilizes full degrees of freedom of single-qubit gates and therefore can potentially obtain better performance. Specifically, it simultaneously optimizes both the axis and the angle of a single-qubit gate, while the known methods either optimize the angle with the axis fixed, or vice versa. It generalizes the known methods and utilizes sinusoidal cost functions parametrized by the axis and angle of rotation. Furthermore, we demonstrate how it can be extended to optimize a set of parametrized two-qubit gates with excitation-conservation constraints, which includes the HOP and the reconfigurable beam-splitter gates. We perform numerical experiments showing the power of the proposed method to find ground states of typical Hamiltonians with quantum imaginary-time evolution using parametrized quantum circuits. In addition, we show the method can be applied to real-time evolution and discuss the tradeoff between its simulation accuracy and hardware efficiency.

Development of variational quantum deep neural networks for image recognition

Parametrized quantum circuits are widely used for supervised learning tasks such as image classification in the noisy intermediate scale quantum era. However, normally, it can only handle low-dimensional data. This study presented a variational quantum deep neural network (VQDNN) model for various scale image recognition tasks. Three classifiers were designed to verify the classification performance of the proposed VQDNN model. In the first classifier, to accommodate the limitations of qubits in both simulation hardware and real quantum hardware – we adopted hybrid principal component analysis – VQDNN architecture. Moreover, the amplitude encoding scheme and the rotation angle coding scheme were employed in the subsequent two classifiers to handle large-size images. Finally, we used the classical neural network and VQDNN model to conduct a comparative experiment of the ten-label classification learning task on the same dataset. The quantum numerical experiment was implemented on two benchmark datasets: the MNIST and UCI databases of handwritten digits. The simulation results showed that the proposed VQDNN classified the two datasets with an accuracy of 100% for the two-class classification task, while the UCI dataset has an accuracy of 90.87% for the ten-label classification task. The proposed VQDNN achieved better classification accuracy than the original classical neural network even under a limited number of qubits available in current hardware, indicating the promising application potential of VQDNN in image recognition.

Quantum agents in the Gym: a variational quantum algorithm for deep Q-learning

Quantum machine learning (QML) has been identified as one of the key fields that could reap advantages from near-term quantum devices, next to optimization and quantum chemistry. Research in this area has focused primarily on variational quantum algorithms (VQAs), and several proposals to enhance supervised, unsupervised and reinforcement learning (RL) algorithms with VQAs have been put forward. Out of the three, RL is the least studied and it is still an open question whether VQAs can be competitive with state-of-the-art classical algorithms based on neural networks (NNs) even on simple benchmark tasks. In this work, we introduce a training method for parametrized quantum circuits (PQCs) that can be used to solve RL tasks for discrete and continuous state spaces based on the deep Q-learning algorithm. We investigate which architectural choices for quantum Q-learning agents are most important for successfully solving certain types of environments by performing ablation studies for a number of different data encoding and readout strategies. We provide insight into why the performance of a VQA-based Q-learning algorithm crucially depends on the observables of the quantum model and show how to choose suitable observables based on the learning task at hand. To compare our model against the classical DQN algorithm, we perform an extensive hyperparameter search of PQCs and NNs with varying numbers of parameters. We confirm that similar to results in classical literature, the architectural choices and hyperparameters contribute more to the agents' success in a RL setting than the number of parameters used in the model. Finally, we show when recent separation results between classical and quantum agents for policy gradient RL can be extended to inferring optimal Q-values in restricted families of environments.

Mode connectivity in the loss landscape of parameterized quantum circuits

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 Goodfellow et al. (2014); Li et al. (2017) and tests for connectivity used in Draxler et al. (2018) to study the loss landscape features in PQC training. We present results for PQCs trained on a simple regression task, using the bilayer circuit ansatz, which consists of alternating layers of parameterized rotation gates and entangling gates. Multiple circuits are trained with 3 different batch gradient optimizers: stochastic gradient descent, the quantum natural gradient, and Adam. We identify large features in the landscape that can lead to faster convergence in training workflows.

Single-component gradient rules for variational quantum algorithms

Many near-term quantum computing algorithms are conceived as variational quantum algorithms, in which parameterized quantum circuits are optimized in a hybrid quantum–classical setup. Examples are variational quantum eigensolvers, quantum approximate optimization algorithms as well as various algorithms in the context of quantum-assisted machine learning. A common bottleneck of any such algorithm is constituted by the optimization of the variational parameters. A popular set of optimization methods work on the estimate of the gradient, obtained by means of circuit evaluations. We will refer to the way in which one can combine these circuit evaluations as gradient rules. This work provides a comprehensive picture of the family of gradient rules that vary parameters of quantum gates individually. The most prominent known members of this family are the parameter shift rule (PSR) and the finite differences method. To unite this family, we propose a generalized PSR that expresses all members of the aforementioned family as special cases, and discuss how all of these can be seen as providing access to a linear combination of exact first- and second-order derivatives. We further prove that a PSR with one non-shifted evaluation and only one shifted circuit evaluation does not exist, and introduce a novel perspective for approaching new gradient rules.

Unsupervised strategies for identifying optimal parameters in Quantum Approximate Optimization Algorithm

AbstractAs combinatorial optimization is one of the main quantum computing applications, many methods based on parameterized quantum circuits are being developed. In general, a set of parameters are being tweaked to optimize a cost function out of the quantum circuit output. One of these algorithms, the Quantum Approximate Optimization Algorithm stands out as a promising approach to tackling combinatorial problems. However, finding the appropriate parameters is a difficult task. Although QAOA exhibits concentration properties, they can depend on instances characteristics that may not be easy to identify, but may nonetheless offer useful information to find good parameters. In this work, we study unsupervised Machine Learning approaches for setting these parameters without optimization. We perform clustering with the angle values but also instances encodings (using instance features or the output of a variational graph autoencoder), and compare different approaches. These angle-finding strategies can be used to reduce calls to quantum circuits when leveraging QAOA as a subroutine. We showcase them within Recursive-QAOA up to depth 3 where the number of QAOA parameters used per iteration is limited to 3, achieving a median approximation ratio of 0.94 for MaxCut over 200 Erdős-Rényi graphs. We obtain similar performances to the case where we extensively optimize the angles, hence saving numerous circuit calls.