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 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.
Quantum circuits with gates (local unitaries) respecting a global symmetry have broad applications in quantum information science and related fields, such as condensed-matter theory and quantum thermodynamics. However, despite their widespread use, fundamental properties of such circuits are not well understood. Recently, it was found that generic unitaries respecting a global symmetry cannot be realized, even approximately, using gates that respect the same symmetry. This observation raises important open questions: What unitary transformations can be realized with k -local gates that respect a global symmetry? In other words, in the presence of a global symmetry, how does the locality of interactions constrain the possible time evolution of a composite system? In this work, we address these questions for the case of Abelian (commutative) symmetries and develop constructive methods for synthesizing circuits with such symmetries. Remarkably, as a corollary, we find that, while the locality of interactions still imposes additional constraints on realizable unitaries, certain restrictions observed in the case of non-Abelian symmetries do not apply to circuits with Abelian symmetries. For instance, in circuits with a general non-Abelian symmetry such as SU ( d ) , the unitary realized in a subspace with one irreducible representation (charge) of the symmetry dictates the realized unitaries in multiple other sectors with inequivalent representations of the symmetry. Furthermore, in certain sectors, rather than all unitaries respecting the symmetry, the realizable unitaries are the symplectic or orthogonal subgroups of this group. We prove that none of these restrictions appears in the case of Abelian symmetries. This result suggests that global non-Abelian symmetries may affect the thermalization of quantum systems in ways not possible under Abelian symmetries. Published by the American Physical Society 2024
Abstract According to an elementary result in quantum computing, any unitary transformation on a composite system can be generated using 2-local unitaries, i.e., those that act only on two subsystems. Beside its fundamental importance in quantum computing, this result can also be regarded as a statement about the dynamics of systems with local Hamiltonians: although locality puts various constraints on the short-term dynamics, it does not restrict the possible unitary evolutions that a composite system with a general local Hamiltonian can experience after a sufficiently long time. We ask if such universality remains valid in the presence of conservation laws and global symmetries. In particular, can k-local symmetric unitaries on a composite system generate all symmetric unitaries on that system? Surprisingly, it turns out that the answer is negative in the case of continuous symmetries, such as U(1) and SU(2): generic symmetric unitaries cannot be implemented, even approximately, using local symmetric unitaries. In the context of quantum thermodynamics this means that generic energy-conserving unitary transformations on a composite system cannot be implemented by applying local energy-conserving unitary transformations on the components. We also show how this no-go theorem can be circumvented via catalysis: any globally energy-conserving unitary can be implemented using a sequence of 2-local energy-conserving unitaries, provided that one can use a single ancillary qubit (catalyst).
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.
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.
Universality relies on combining operators with different algebraic properties, which depend on the physical structure of the Hilbert space encoding the information. Systems of identical symmetric particles---including bosons and symmetric collections of spin-1/2 particles---can be mapped to angular-momentum systems, where rotations supplemented by quadratic, spin-squeezing interactions provide universal control. We revisit universality in this framework and confirm that such linear and quadratic operations provide universal control on every fixed-particle-number subspace. At the abstract level, however, we show that the dynamical Lie algebra generated by all degree-at-most-two elements does not exhaust the full universal enveloping algebra of suR(2), contrary to a stronger claim made previously: higher powers of the quadratic Casimir element cannot be generated by commutators and linear combinations. This obstruction does not conflict with fixed-sector universality because, under a fixed irreducible spin-n/2 representation, all powers of the Casimir element act as scalar multiples of the identity. We determine the resulting abstract dynamical Lie algebras exactly and show that the central-free and full degree-at-most-two algebras map, respectively, onto suR(n+1) and uR(n+1), establishing special-unitary and full-unitary universality. We further derive a practical degree-two universality criterion and discuss finite-dimensional SSRC universality.