We develop a theoretical framework for S n -equivariant convolutional quantum circuits with SU(d) symmetry, building on and significantly generalizing Jordan's permutational quantum computing formalism based on Schur-Weyl duality connecting both SU(d) and S n actions on qudits.In particular, we utilize the Okounkov-Vershik approach to prove Harrow's statement on the equivalence between SU(d) and S n irrep bases and to establish the S n -equivariant convolutional quantum alternating ansätze (S n -CQA) using Young-Jucys-Murphy elements.We prove that S n -CQA is able to generate any unitary in any given S n irrep sector, which may serve as a universal model for a wide array of quantum machine-learning problems with the presence of SU(d) symmetry.Our method provides another way to prove the universality of the quantum approximate optimization algorithm and verifies that four-local SU(d)-symmetric unitaries are sufficient to build generic SU(d)-symmetric quantum circuits up to relative phase factors.We present numerical simulations to showcase the effectiveness of the ansätze to find the ground-state energy of the J 1 -J 2 antiferromagnetic Heisenberg model on the rectangular and kagome lattices.Our work provides the first application of the celebrated Okounkov-Vershik S n representation theory to quantum physics and machine learning, from which to propose quantum variational ansätze that strongly suggests to be classically intractable tailored towards a specific optimization problem.
We develop a theoretical framework for S n -equivariant convolutional quantum circuits with SU(d) symmetry, building on and significantly generalizing Jordan's permutational quantum computing formalism based on Schur-Weyl duality connecting both SU(d) and S n actions on qudits.In particular, we utilize the Okounkov-Vershik approach to prove Harrow's statement on the equivalence between SU(d) and S n irrep bases and to establish the S n -equivariant convolutional quantum alternating ansätze (S n -CQA) using Young-Jucys-Murphy elements.We prove that S n -CQA is able to generate any unitary in any given S n irrep sector, which may serve as a universal model for a wide array of quantum machine-learning problems with the presence of SU(d) symmetry.Our method provides another way to prove the universality of the quantum approximate optimization algorithm and verifies that four-local SU(d)-symmetric unitaries are sufficient to build generic SU(d)-symmetric quantum circuits up to relative phase factors.We present numerical simulations to showcase the effectiveness of the ansätze to find the ground-state energy of the J 1 -J 2 antiferromagnetic Heisenberg model on the rectangular and kagome lattices.Our work provides the first application of the celebrated Okounkov-Vershik S n representation theory to quantum physics and machine learning, from which to propose quantum variational ansätze that strongly suggests to be classically intractable tailored towards a specific optimization problem.
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 generation of k -designs (pseudorandom distributions that emulate the Haar measure up to k moments) with local quantum circuit ensembles is a problem of fundamental importance in quantum information and physics. Despite the extensive understanding of this problem for ordinary random circuits, the crucial situations in which symmetries or conservation laws are in play are known to pose fundamental challenges and remain little understood. Here, we construct explicit local unitary ensembles that can achieve high-order unitary k -designs under transversal continuous symmetry, in the particularly important SU ( d ) case. Specifically, we define the convolutional quantum alternating (CQA) group generated by 4-local SU ( d ) -symmetric Hamiltonians as well as associated 4-local SU ( d ) -symmetric random unitary circuit ensembles and prove that they form and converge to SU ( d ) -symmetric k -designs, respectively, for all k < n ( n − 3 ) / 2 , with n being the number of qudits. A key technique that we employ to obtain the results is the Okounkov-Vershik approach to S n representation theory. To study the convergence time of the CQA ensemble, we develop a numerical method using the Young orthogonal form and the S n branching rule. We provide strong evidence for a subconstant spectral gap and certain convergence time scales of various important circuit architectures, which contrast with the symmetry-free case. We also provide comprehensive explanations of the difficulties and limitations in rigorously analyzing the convergence time using methods that have been effective for cases without symmetries, including Knabe’s local gap threshold and Nachtergaele’s martingale methods. This suggests that a novel approach is likely necessary for understanding the convergence time of SU ( d ) -symmetric local random circuits. Published by the American Physical Society 2024
We introduce a framework of the equivariant convolutional quantum algorithms which is tailored for a number of machine-learning tasks on physical systems with arbitrary SU(d) symmetries. It allows us to enhance a natural model of quantum computation---permutational quantum computing (PQC) [Jordan, Quantum Inf. Comput. 10, 470 (2010)]---and define a more powerful model: PQC+. While PQC was shown to be efficiently classically simulatable, we exhibit a problem which can be efficiently solved on a PQC+ machine, whereas no classical polynomial time algorithm is known, thus providing evidence against PQC+ being classically simulatable. We further discuss practical quantum machine learning algorithms which can be carried out in the paradigm of PQC+.
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.
Quantum neural network architectures that have little to no inductive biases are known to face trainability and generalization issues. Inspired by a similar problem, recent breakthroughs in machine learning address this challenge by creating models encoding the symmetries of the learning task. This is materialized through the usage of equivariant neural networks the action of which commutes with that of the symmetry. In this work, we import these ideas to the quantum realm by presenting a comprehensive theoretical framework to design equivariant quantum neural networks (EQNNs) for essentially any relevant symmetry group. We develop multiple methods to construct equivariant layers for EQNNs and analyze their advantages and drawbacks. Our methods can find unitary or general equivariant quantum channels efficiently even when the symmetry group is exponentially large or continuous. As a special implementation, we show how standard quantum convolutional neural networks (QCNNs) can be generalized to group-equivariant QCNNs where both the convolution and pooling layers are equivariant to the symmetry group. We then numerically demonstrate the effectiveness of a S U ( 2 ) -equivariant QCNN over symmetry-agnostic QCNN on a classification task of phases of matter in the bond-alternating Heisenberg model. Our framework can be readily applied to virtually all areas of quantum machine learning. Lastly, we discuss about how symmetry-informed models such as EQNNs provide hopes to alleviate central challenges such as barren plateaus, poor local minima, and sample complexity. Published by the American Physical Society 2024
We propose SnCQA, a set of hardware-efficient variational circuits of equivariant quantum convolutional circuits respective to permutation symmetries and spatial lattice symmetries with the number of qubits n. By exploiting permutation symmetries of the system, such as lattice Hamiltonians common to many quantum many-body and quantum chemistry problems, Our quantum neural networks are suitable for solving machine learning problems where permutation symmetries are present, which could lead to significant savings of computational costs. Aside from its theoretical novelty, we find our simulations perform well in practical instances of learning ground states in quantum computational chemistry, where we could achieve comparable performances to traditional methods with few tens of parameters. Compared to other traditional variational quantum circuits, such as the pure hardware-efficient ansatz (pHEA), we show that SnCQA is more scalable, accurate, and noise resilient (with 20× better performance on 3 × 4 square lattice and 200% - 1000% resource savings in various lattice sizes and key criterions such as the number of layers, parameters, and times to converge in our cases), suggesting a potentially favorable experiment on near-time quantum devices.