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Physical Review A

Subspace controllability of symmetric spin networks

Jiahui Chen, Yehao Zhou, Ji Bian, Jun Li, Xinhua Peng

Published Sep 3, 2020• 7 citations
Publisher

Abstract

Controlling the dynamics of quantum systems has been a long-standing goal of quantum physics. To understand the controllability of a finite-dimensional quantum system is a fundamental problem in quantum control theory, but can be very challenging in many cases. In particular, results on controllability of open-loop coherently controlled closed quantum systems under symmetry constraints have been elusive. In this paper, we study the controllability of spin systems with some specific symmetry where spins are coupled via Ising interaction and manipulated collectively by a global control field. Based on Lie group and Lie algebra techniques, we find the conditions under which the system is universally controllable in its irreducible subspaces. We believe that the results obtained here will contribute to quantum controlling on symmetric quantum systems.

DOI
10.1103/physreva.102.032602

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Abstract. This paper describes a framework for the controllability analysis of networks of [Formula: see text] quantum systems of an arbitrary dimension [Formula: see text] , qudits, with dynamics determined by Hamiltonians that are invariant under the permutation group [Formula: see text]. Because of the symmetry, the underlying Hilbert space, [Formula: see text], splits into invariant subspaces for the Lie algebra of [Formula: see text]-invariant elements in [Formula: see text], denoted here by [Formula: see text]. The dynamical Lie algebra [Formula: see text], which determines the controllability properties of the system, is a Lie subalgebra of such a Lie algebra [Formula: see text]. If [Formula: see text] acts as [Formula: see text] on each of the invariant subspaces [Formula: see text], the system is called subspace controllable. The described approach is based on recognizing that such a splitting of the Hilbert space [Formula: see text] coincides with the Clebsch–Gordan splitting of [Formula: see text] into irreducible representations of [Formula: see text]. In this view, [Formula: see text] is the direct sum of certain [Formula: see text] for some [Formula: see text]’s we shall specify and its center, which is the Abelian (Lie) algebra generated by the Casimir operators. Most of the results presented are for general [Formula: see text] and [Formula: see text], but the paper provides a complete treatment and proof of subspace controllability for the new case of a system with [Formula: see text], [Formula: see text], that is, three qutrits. The results are motivated by recent great interest in symmetric quantum states and systems (see, e.g., [K. Eckert et al., Ann. Phys., 299 (2001), pp. 88–127], [A. W. Harrow, “The Church of the Symmetric Subspace,” https://arxiv.org/abs/1308.6595 , 2013], [P. Migdal, J. Rodriguez-Laguna, and M. Lewenstein, Phys. Rev. A, 88 (2013), 021335]) both at the theoretical and experimental levels and by recent proposals in geometric quantum machine learning [M. Cerezo et al., Nature Comput. Sci., 2 (2022), pp. 567–576], [Q. T. Nguyen et al., PRX Quant., 5 (2024), 020328] to exploit symmetries in the data and in quantum circuits to improve the performance of learning protocols.

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Controlling the dynamics of quantum systems has been a long-standing goal of quantum physics. To understand the controllability of a finite-dimensional quantum system is a fundamental problem in quantum control theory, but can be very challenging in many cases. In particular, results on controllability of open-loop coherently controlled closed quantum systems under symmetry constraints have been elusive. In this paper, we study the controllability of spin systems with some specific symmetry where spins are coupled via Ising interaction and manipulated collectively by a global control field. Based on Lie group and Lie algebra techniques, we find the conditions under which the system is universally controllable in its irreducible subspaces. We believe that the results obtained here will contribute to quantum controlling on symmetric quantum systems.
We consider a class of spin networks where each spin in a certain set interacts, via Ising coupling, with a set of central spins, and the control acts simultaneously on all the spins. This is a common situation for instance in NV centers in diamonds, and we focus on the physical case of up to two central spins. Due to the permutation symmetries of the network, the system is not globally controllable but it displays invariant subspaces of the underlying Hilbert space. The system is said to be subspace controllable if it is controllable on each of these subspaces. We characterize the given invariant subspaces and the dynamical Lie algebra of this class of systems and prove subspace controllability in every case.
We consider a class of spin networks where each spin in a certain set interacts via Ising coupling with a single central spin. This is a common situation for instance in NV centers in diamonds. Due to the permutation symmetries of the network, the system is not globally controllable but it displays invariant subspaces of the underlying Hilbert space. The system is said to be subspace controllable if it is controllable on each of these subspaces. We characterize the given invariant subspaces and the dynamical Lie algebra of this class of systems and prove subspace controllability.
We describe a framework for the controllability analysis of networks of nnn quantum systems of an arbitrary dimension ddd, {\it qudits}, with dynamics determined by Hamiltonians that are invariant under the permutation group SnS_nSn​. Because of the symmetry, the underlying Hilbert space, H=(Cd)⊗n{\cal H}=(\mathbb{C}^d)^{\otimes n}H=(Cd)⊗n, splits into invariant subspaces for the Lie algebra of SnS_nSn​-invariant elements in u(dn)u(d^n)u(dn), denoted here by uSn(dn)u^{S_n}(d^n)uSn​(dn). The dynamical Lie algebra L{\cal L}L, which determines the controllability properties of the system, is a Lie subalgebra of such a Lie algebra uSn(dn)u^{S_n}(d^n)uSn​(dn). If L{\cal L}L acts as su(dim⁡(V))su\left( \dim(V) \right)su(dim(V)) on each of the invariant subspaces VVV, the system is called {\it subspace controllable}. Our approach is based on recognizing that such a splitting of the Hilbert space H{\cal H}H coincides with the {\it Clebsch-Gordan} splitting of (Cd)⊗n(\mathbb{C}^d)^{\otimes n}(Cd)⊗n into {\it irreducible representations} of su(d)su(d)su(d). In this view, uSn(dn)u^{S_n}(d^n)uSn​(dn), is the direct sum of certain su(nj)su(n_j)su(nj​) for some njn_jnj​'s we shall specify, and its {\it center} which is the Abelian (Lie) algebra generated by the {\it Casimir operators}. Generalizing the situation previously considered in the literature, we consider dynamics with arbitrary local simultaneous control on the qudits and a symmetric two body interaction. Most of the results presented are for general nnn and ddd but we recast previous results on nnn qubits in this new general framework and provide a complete treatment and proof of subspace controllability for the new case of n=3n=3n=3, d=3d=3d=3, that is, {\it three qutrits}.
Abstract. This paper describes a framework for the controllability analysis of networks of [Formula: see text] quantum systems of an arbitrary dimension [Formula: see text] , qudits, with dynamics determined by Hamiltonians that are invariant under the permutation group [Formula: see text]. Because of the symmetry, the underlying Hilbert space, [Formula: see text], splits into invariant subspaces for the Lie algebra of [Formula: see text]-invariant elements in [Formula: see text], denoted here by [Formula: see text]. The dynamical Lie algebra [Formula: see text], which determines the controllability properties of the system, is a Lie subalgebra of such a Lie algebra [Formula: see text]. If [Formula: see text] acts as [Formula: see text] on each of the invariant subspaces [Formula: see text], the system is called subspace controllable. The described approach is based on recognizing that such a splitting of the Hilbert space [Formula: see text] coincides with the Clebsch–Gordan splitting of [Formula: see text] into irreducible representations of [Formula: see text]. In this view, [Formula: see text] is the direct sum of certain [Formula: see text] for some [Formula: see text]’s we shall specify and its center, which is the Abelian (Lie) algebra generated by the Casimir operators. Most of the results presented are for general [Formula: see text] and [Formula: see text], but the paper provides a complete treatment and proof of subspace controllability for the new case of a system with [Formula: see text], [Formula: see text], that is, three qutrits. The results are motivated by recent great interest in symmetric quantum states and systems (see, e.g., [K. Eckert et al., Ann. Phys., 299 (2001), pp. 88–127], [A. W. Harrow, “The Church of the Symmetric Subspace,” https://arxiv.org/abs/1308.6595 , 2013], [P. Migdal, J. Rodriguez-Laguna, and M. Lewenstein, Phys. Rev. A, 88 (2013), 021335]) both at the theoretical and experimental levels and by recent proposals in geometric quantum machine learning [M. Cerezo et al., Nature Comput. Sci., 2 (2022), pp. 567–576], [Q. T. Nguyen et al., PRX Quant., 5 (2024), 020328] to exploit symmetries in the data and in quantum circuits to improve the performance of learning protocols.
Global control offers a promising route to scalable quantum computing. A recent conjecture by Hu et al. (arXiv:2508.19075) proposes that any connected qubit graph equipped with global Ising-type interactions and tunable global transverse fields achieves universality if and only if an additional control field breaks every non-trivial automorphism of the underlying graph. We disprove this conjecture by exhibiting explicit seven- and nine-qubit counterexamples: connected graphs with trivial automorphism group for which the generated Lie algebra is nonetheless not universal. Our analysis reveals that graph automorphisms capture only part of the Hamiltonian symmetry structure: there exist hidden symmetries beyond the automorphism group of the graph. Additionally, in the case of non-trivial automorphism group, we find control terms which break the graph symmetries but are still not universal. These findings sharpen the characterization of universality for globally controlled quantum systems.
In this paper, we present a controllability analysis of the quantum Ising periodic chain of n spin 1/2 particles where the interpolating parameter between the two Hamiltonians plays the role of the control. A fundamental result in the control theory of quantum systems states that the set of achievable evolutions is (dense in) the Lie group corresponding to the Lie algebra generated by the Hamiltonians of the system. Such a dynamical Lie algebra therefore characterizes all the state transitions available for a given system. For the Ising spin periodic chain we characterize such a dynamical Lie algebra and therefore the set of all reachable states. In particular, we prove that the dynamical Lie algebra is a (3n-1)-dimensional Lie sub-algebra of su(2^n) which is a direct sum of a two dimensional center and a (3n-3)-dimensional semisimple Lie subalgebra. This in turn is the direct sum of n-1 Lie algebras isomorphic to su(2) parametrized by the eigenvalues of a fixed matrix. We display the basis for each of these Lie subalgebras. Therefore the problem of control for the Ising spin periodic chain is, modulo the two dimensional center, a problem of simultaneous control of n-1 spin 1/2 particles. In the process of proving this result, we develop some tools which are of general interest for the controllability analysis of quantum systems with symmetry.