Abstract Practical success of quantum learning models hinges on having a suitable structure for the parameterized quantum circuit. Such structure is defined both by the types of gates employed and by the correlations of their parameters. While much research has been devoted to devising adequate gate-sets, typically respecting some symmetries of the problem, very little is known about how their parameters should be structured. In this work, we show that an ideal parameter structure naturally emerges when carefully considering spatial symmetries (i.e. the symmetries that are permutations of parts of the system under study). Namely, we consider the automorphism group of the problem Hamiltonian, leading us to develop a circuit construction that is equivariant under this symmetry group. The benefits of our novel circuitstructure, called ORB, are numerically probed in several ground-state problems. We find a consistent improvement (in terms of circuit depth, number of parameters required, and gradient magnitudes) compared to literature circuit constructions.
Abstract Practical success of quantum learning models hinges on having a suitable structure for the parameterized quantum circuit. Such structure is defined both by the types of gates employed and by the correlations of their parameters. While much research has been devoted to devising adequate gate-sets, typically respecting some symmetries of the problem, very little is known about how their parameters should be structured. In this work, we show that an ideal parameter structure naturally emerges when carefully considering spatial symmetries (i.e. the symmetries that are permutations of parts of the system under study). Namely, we consider the automorphism group of the problem Hamiltonian, leading us to develop a circuit construction that is equivariant under this symmetry group. The benefits of our novel circuitstructure, called ORB, are numerically probed in several ground-state problems. We find a consistent improvement (in terms of circuit depth, number of parameters required, and gradient magnitudes) compared to literature circuit constructions.
Abstract Variational algorithms require architectures that naturally constrain the optimization space to run efficiently. Geometric quantum machine learning achieves this goal by encoding group structure into parameterized quantum circuits to include the symmetries of a problem as an inductive bias. However, constructing such circuits is challenging as a concrete guiding principle has yet to emerge. In this paper, we propose the use of spin networks , a form of directed tensor network invariant under a group transformation, to devise SU(2) equivariant quantum circuit ansätze—circuits possessing spin-rotation symmetry. By changing to the basis that block diagonalizes the SU(2) group action, these networks provide a natural building block for constructing parameterized equivariant quantum circuits. We prove that our construction is mathematically equivalent to other known constructions, such as those based on twirling and generalized permutations, but more direct to implement on quantum hardware. The efficacy of our constructed circuits is tested by solving the ground state problem of SU(2) symmetric Heisenberg models on the one-dimensional triangular lattice and the Kagome lattice. Our results highlight that our equivariant circuits boost the performance of quantum variational algorithms, indicating broader applicability to other real-world problems.
The success of variational quantum learning models crucially depends on choosing parametrizations that reflect the structure of the problem at hand. Symmetries provide one of the clearest such structures: whenever transformations of the input leave the desired outcome unchanged, this invariance should be built into the model rather than discovered during training. However, imposing a symmetry does not by itself determine a useful ansatz. Even within the symmetry-preserving space, one must decide where the trainable degrees of freedom should be placed. In this work, we study this remaining design freedom in equivariant variational quantum circuits. Building on symmetry-based parameter sharing, we disentangle two architectural choices: how much symmetry should be enforced, and which symmetry-respecting interactions should be trainable. Using Tic-Tac-Toe as a fully enumerable and structurally transparent test case, we find that suitable subgroups preserve most of the generalization benefit. By contrast, the dominant gains arise from gates acting directly on decisive task motifs. Thus, symmetry defines the admissible design space, while effective ansätze require an additional task-informed choice of trainable interactions.
The variational quantum simulation technique on gate-based quantum processors can be made more efficient if symmetries of the Hamiltonian system of interest are factored into the design construction of the underlying parameterized quantum circuits (PQC) needed by the technique. In the first part of this work, we present an easy and practical approach that can be used to construct symmetric quantum circuits. The method we present can directly be adapted for any symmetry, though in the present work we focus explicitly on the examples of Z 2 , Z 3 , and particle number conservation. We show how to construct symmetric gates that respects the aforementioned symmetries. In the second part, we focus on numerical analysis of parameterized particle-conserving gates. In variational quantum simulation, for example, the variational quantum eigensolver (VQE), the symmetric gates of the PQC are parameterized. Since there are often many possible inequivalent parametrizations, it is often not immediately obvious which gate parametrization would be most efficient in general for a wide range of Hamiltonians that respect the symmetry. We streamline our analysis to particle-conserving circuits and Hamiltonians. We numerically analyze the efficiency of different particle-conserving gate parametrizations by studying two examples, namely, a physics problem of learning the ground state of Heisenberg spin chains, and an artificial problem of learning the random arbitary states sampled within the particle number conserved subspace.
In the current framework of geometric quantum machine learning, the canonical method for constructing a variational ansatz that respects the symmetry of some group action is by forcing the circuit to be equivariant, i.e., to commute with the action of the group. This can, however, be an overzealous constraint that greatly limits the expressivity of the circuit, especially in the case of continuous symmetries. We propose an alternative paradigm for the symmetry-informed construction of variational quantum circuits, based on homogeneous spaces, relaxing the overly stringent requirement of equivariance. We achieve this by introducing horizontal quantum gates, which only transform the state with respect to the directions orthogonal to those of the symmetry. We show that horizontal quantum gates are much more expressive than equivariant gates, and thus can solve problems that equivariant circuits cannot. For instance, a circuit comprised of horizontal gates can find the ground state of an SU ( 2 ) -symmetric model where the ground state spin sector is unknown—a task where equivariant circuits fall short. Moreover, for a particular subclass of horizontal gates based on symmetric spaces, we can obtain efficient circuit decompositions for our gates through the KAK theorem. Finally, we highlight a particular class of horizontal quantum gates that behave similarly to general SU ( 4 ) gates, while achieving a quadratic reduction in the number of parameters for a generic problem.
Abstract Variational quantum algorithms are the leading candidate for advantage on near-term quantum hardware. When training a parametrized quantum circuit in this setting to solve a specific problem, the choice of ansatz is one of the most important factors that determines the trainability and performance of the algorithm. In quantum machine learning (QML), however, the literature on ansatzes that are motivated by the training data structure is scarce. In this work, we introduce an ansatz for learning tasks on weighted graphs that respects an important graph symmetry, namely equivariance under node permutations. We evaluate the performance of this ansatz on a complex learning task, namely neural combinatorial optimization, where a machine learning model is used to learn a heuristic for a combinatorial optimization problem. We analytically and numerically study the performance of our model, and our results strengthen the notion that symmetry-preserving ansatzes are a key to success in QML.
A quantum circuit learning approach is studied to carry out the fast-fitting of Gaussian functions. First, a parameterized structure is designed for quantum circuits based on the boson sampling model. And then, the training procedure of exploiting gradient-based optimizations is presented to iteratively update the gradient of the loss function concerning circuit parameters. For efficiency, two kinds of circuit loss, the kernel maximum mean discrepancy and the mean absolute error, are used in the training procedure, which are both competent to achieve quantum circuit learning well. It is significant that the two circuit losses assist in reducing the variance to2.54×10−6and6.91×10−6, respectively. Finally, a kind of quantum circuit fixed structure is developed with the boson sampling model that can decrease the model complexity as the circuit depth d grows. Sets of experiments have been conducted to evaluate the proposed quantum circuit learning scheme, and demonstrate that our parameterized approach is efficient and promising, and it is worth looking forward to solving practical application problems with quantum computers since valid quantum circuits for Gaussian function fast-fitting can be designed indeed.