We describe the implications of permutation symmetry for the state space and dynamics of quantum mechanical systems of matrices of general size
N. We solve the general 11-parameter permutation invariant quantum matrix harmonic oscillator Hamiltonian and calculate the canonical partition function. The permutation invariant sector of the Hilbert space, for general Hamiltonians, can be described using partition algebra diagrams forming the bases of a tower of partition algebras
Pk(N). The integer
k is interpreted as the degree of matrix oscillator polynomials in the quantum mechanics. Families of interacting Hamiltonians are described which are diagonalized by a representation theoretic basis for the permutation invariant subspace which we construct for $N\ensuremath{\ge}2k$. These include Hamiltonians for which the low-energy states are permutation invariant and can give rise to large ground state degeneracies related to the dimensions of partition algebras. A symmetry-based mechanism for quantum many body scars discussed in the literature can be realized in these matrix systems with permutation symmetry. A mapping of the matrix index values to lattice sites allows a realization of the mechanism in the context of modified Bose-Hubbard models. Extremal correlators analogous to those studied in
AdS/CFT are shown to obey selection rules based on Clebsch-Gordan multiplicites (Kronecker coefficients) of symmetric groups.
We introduce a practical construction of group-equivariant and permutation-invariant functions of
N variables given a finite-dimensional space stable with respect to the group action. The construction applies to any connected linear Lie group and relies on leveraging the Lie algebra to build a matrix
M whose kernel is in one-to-one correspondence with the subspace with desired equivariance and invariance properties, removing the need for prior knowledge of Clebsch--Gordan coefficients. A similar construction is proposed for group-equivariant functions alone, without imposing permutation-invariance. For the groups
SO(3) and
SU(2), we further exploit the structure of the Lie algebra to demonstrate the sparsity pattern and rank of the matrix
M, which yields the exact dimension of the group-equivariant and permutation-invariant space, as well as the dimension of the group-equivariant space alone. We demonstrate analytically and verify numerically that the proposed method scales linearly with respect to the dimensionality of the basis, offering a high computational gain compared to existing methods in the literature which typically scale exponentially. We finally perform a dimensionality comparison, showing that for large values of~
N, the dimension of group-equivariant and permutation-invariant spaces is of comparable order as the dimension of permutation-invariant spaces, while pre-asymptotically, the first dimensionality is orders of magnitude lower than the second. Hence a substantial computational gain can be achieved by explicitly enforcing group-equivariance on top of permutation-invariance when approximating such functions.
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.
We prove a generalization of the quantum de Finetti theorem when the local space is an infinite-dimensional Fock space. In particular, instead of considering the action of the permutation group on n copies of that space, we consider the action of the unitary group U(n) on the creation operators of the n modes and define a natural generalization of the symmetric subspace as the space of states invariant under unitaries in U(n). Our first result is a complete characterization of this subspace, which turns out to be spanned by a family of generalized coherent states related to the special unitary group SU(p, q) of signature (p, q). More precisely, this construction yields a unitary representation of the noncompact simple real Lie group SU(p, q). We therefore find a dual unitary representation of the pair of groups U(n) and SU(p, q) on an n(p + q)-mode Fock space. The (Gaussian) SU(p, q) coherent states resolve the identity on the symmetric subspace, which implies a Gaussian de Finetti theorem stating that tracing over a few modes of a unitary-invariant state yields a state close to a mixture of Gaussian states. As an application of this de Finetti theorem, we show that the n × n upper-left submatrix of an n × n Haar-invariant unitary matrix is close in total variation distance to a matrix of independent normal variables if n3 = O(m).
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.
We prove a generalization of the quantum de Finetti theorem when the local space is an infinite-dimensional Fock space. In particular, instead of considering the action of the permutation group on n copies of that space, we consider the action of the unitary group U(n) on the creation operators of the n modes and define a natural generalization of the symmetric subspace as the space of states invariant under unitaries in U(n). Our first result is a complete characterization of this subspace, which turns out to be spanned by a family of generalized coherent states related to the special unitary group SU(p,q) of signature (p,q). More precisely, this construction yields a unitary representation of the noncompact simple real Lie group SU(p,q). We therefore find a dual unitary representation of the pair of groups U(n) and SU(p,q) on an n(p+q)-mode Fock space.The (Gaussian) SU(p,q) coherent states resolve the identity on the symmetric subspace, which implies a Gaussian de Finetti theorem stating that tracing over a few modes of a unitary-invariant state yields a state close to a mixture of Gaussian states. As an application of this de Finetti theorem, we show that the n×n upper-left submatrix of an n×n Haar-invariant unitary matrix is close in total variation distance to a matrix of independent normal variables if n3=O(m).