Symmetry-restricted quantum circuits are still well-behaved
Maximilian Balthasar Mansky, Santiago Londoño Castillo, Miguel Armayor Martinez, Alejandro Bravo de la Serna, Gautham Sathish, Zhihao Wang, Sebastian Wölckerlt, Claudia Linnhoff–Popien
We show that quantum circuits restricted by a symmetry inherit the properties of the whole special unitary group $SU(2^n)$, in particular composition, algebraic and topological closedness and connectedness. It extends prior work on symmetric states to the operators and shows that the operator space follows the same structure as the state space. The well-behavedness is independent of the symmetry requirement imposed on the subgroup. We provide an example of a permutation invariance across all qubits.
We show that quantum circuits restricted by a symmetry inherit the properties of the whole special unitary group SU(2n), in particular composition, algebraic and topological closedness and connectedness. It extends prior work on symmetric states to the operators and shows that the operator space follows the same structure as the state space. The well-behavedness is independent of the symmetry requirement imposed on the subgroup. We provide an example of a permutation invariance across all qubits.
Abstract Symmetries are omnipresent in physics and have been used to reduce the number of degrees of freedom of systems. In this work, we investigate the properties of M SU ( n ) , M -invariant subspaces of the special unitary Lie group SU(n). This group is relevant to quantum computing and quantum systems in general. We demonstrate that for certain choices of M , the subset M SU ( n ) inherits many topological and group properties from SU(n). We then present a combinatorial method for computing the dimension of such subspaces when M is a representation of a permutation group acting on a tensor product of spaces G SU ( n ) , or a Hamiltonian H ( n ) SU ( n ) . The Kronecker product of su ( 2 ) matrices is employed to construct the Lie algebras associated with different permutation-invariant groups GSU(n). Numerical results on the number of dimensions support the developed theory.
The implementation of physical symmetries into problem descriptions allows for the reduction of parameters and computational complexity. We show the integration of the permutation symmetry as the most restrictive discrete symmetry into quantum circuits. The permutation symmetry is the supergroup of all other discrete groups. We identify the permutation with a SWAP operation on the qubits. Based on the extension of the symmetry into the corresponding Lie algebra, quantum circuit element construction is shown via exponentiation. This allows for ready integration of the permutation group symmetry into quantum circuit ansatzes. The scaling of the number of parameters is found to be O(n3), significantly lower than the general case and an indication that symmetry restricts the applicability of quantum computing. We also show how to adapt existing circuits to be invariant under a permutation symmetry by modification.
An important result in the theory of quantum control is the "universality" of 2-local unitary gates, i.e. the fact that any global unitary evolution of a system of L qudits can be implemented by composition of 2-local unitary gates. Surprisingly, recent results have shown that universality can break down in the presence of symmetries: in general, not all globally symmetric unitaries can be constructed using k-local symmetric unitary gates. This also restricts the dynamics that can be implemented by symmetric local Hamiltonians. In this paper, we show that obstructions to universality in such settings can in general be understood in terms of superoperator symmetries associated with unitary evolution by restricted sets of gates. These superoperator symmetries lead to block decompositions of the operator Hilbert space, which dictate the connectivity of operator space, and hence the structure of the dynamical Lie algebra. We demonstrate this explicitly in several examples by systematically deriving the superoperator symmetries from the gate structure using the framework of commutant algebras, which has been used to systematically derive symmetries in other quantum many-body systems. We clearly delineate two different types of non-universality, which stem from different structures of the superoperator symmetries, and discuss its signatures in physical observables. In all, our work establishes a comprehensive framework to explore the universality of unitary circuits and derive physical consequences of its absence.
An important result in the theory of quantum control is the “universality” of 2-local unitary gates, i.e., the fact that any global unitary evolution of a system of L qudits can be implemented by composition of 2-local unitary gates. Surprisingly, recent results have shown that universality can break down in the presence of symmetries: in general, not all globally symmetric unitaries can be constructed using k -local symmetric unitary gates. This also restricts the dynamics that can be implemented by symmetric local Hamiltonians. In this paper, we show that obstructions to universality in such settings can in general be understood in terms of superoperator symmetries associated with unitary evolution by restricted sets of gates. These superoperator symmetries lead to block decompositions of the operator Hilbert space, which dictate the connectivity of operator space, and hence the structure of the dynamical Lie algebra. We demonstrate this explicitly in several examples by systematically deriving the superoperator symmetries from the gate structure using the framework of commutant algebras, which has been used to systematically derive symmetries in other quantum many-body systems. We clearly delineate two different types of nonuniversality, which stem from different structures of the superoperator symmetries, and discuss its signatures in physical observables. In all, our work establishes a comprehensive framework to explore the universality of unitary circuits and derive physical consequences of its absence.
Local symmetric quantum circuits are those that contain only local unitaries that respect a certain symmetry. These circuits provide a simple framework to study the dynamics and phases of complex quantum systems with conserved charges; however, some of their basic properties have not yet been understood. Recently, it has been shown that such quantum circuits only generate a restricted subset of symmetric unitary transformations. In this paper, we consider circuits with 2-local SU(d)-invariant unitaries acting on qudits, i.e., d-dimensional quantum systems. Our results reveal a significant distinction between the cases of d=2 and d>=3. For qubits with SU(2) symmetry, arbitrary global rotationally-invariant unitaries can be generated with 2-local ones, up to relative phases between the subspaces corresponding to inequivalent irreducible representations (irreps) of the symmetry, i.e., sectors with different angular momenta. On the other hand, for d>=3, in addition to similar constraints on relative phases between the irreps, locality also restricts the generated unitaries inside these conserved subspaces. In particular, for certain irreps of SU(d), the dynamics under 2-local SU(d)-invariant unitaries can be mapped to the dynamics of a non-interacting (free) fermionic system, whereas for general 3-local ones, the corresponding fermionic model is interacting. Using this correspondence, we obtain new conservation laws for dynamics under 2-local SU(d)-invariant unitaries. Furthermore, we identify a Z2 symmetry related to the parity of permutations which imposes additional conservation laws for systems with n =3. Our results imply that the distribution of unitaries generated by random 2-local SU(d)-invariant unitaries does not converge to the Haar measure over the group of all SU(d)-invariant unitaries, and in fact, for d>=3, is not even a 2-design for the Haar distribution.
Local symmetric quantum circuits provide a simple framework to study the dynamics and phases of complex quantum systems with conserved charges. However, some of their basic properties have not yet been understood. Recently, it has been shown that such quantum circuits only generate a restricted subset of symmetric unitary transformations [I. Marvian, Nature Physics, 2022]. In this paper, we consider circuits with 2-local SU(d)-invariant unitaries acting on qudits, i.e., d-dimensional quantum systems. Our results reveal a significant distinction between the cases of d = 2 and d>2. For qubits with SU(2) symmetry, arbitrary global rotationally-invariant unitaries can be generated with 2-local ones, up to relative phases between the subspaces corresponding to inequivalent irreducible representations (irreps) of the symmetry, i.e., sectors with different angular momenta. On the other hand, for d>2, in addition to similar constraints on the relative phases between the irreps, locality also restricts the generated unitaries inside these conserved subspaces. These constraints impose conservation laws that hold for dynamics under 2-local SU(d)-invariant unitaries, but are violated under general SU(d)-invariant unitaries. Based on this result, we show that the distribution of unitaries generated by random 2-local SU(d)-invariant unitaries does not converge to the Haar measure over the group of all SU(d)-invariant unitaries, and in fact, for d>2, is not even a 2-design for the Haar distribution.