Motivated by the novel applications of the mathematical formalism of quantum theory and its generalizations in cognitive science, psychology, social and political sciences, and economics, we extend the notion of the tensor product and entanglement. We also study the relation between conventional entanglement of complex qubits and our generalized entanglement. Our construction can also be used to describe entanglement in the framework of non-Archimedean physics. It is also possible to construct tensor products of non-Archimedean (e.g., p-adic) and complex Hilbert spaces.
Quantum computations that involve only Clifford operations are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup G of U(d). In particular we seek G such that G tensor G has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case d=2 we exhaustively treat all finite subgroups of U(2) and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups G_n generated by X and the n^th root of Z.
Quantum computations starting with computational basis states and involving only Clifford operations, are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup
G of
U(d). In particular we seek
G such that
G⊗G has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case
d=2 we exhaustively treat all finite irreducible subgroups of
U(2) and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups generated by X and the
nth root of
Z for
n∈N.
Self-dualities and the stacking of symmetry-protected topological (SPT) phases are basic operations on quantum many-body systems. For a
Zp one-form symmetry in 3+1d these correspond to the Kramers-Wannier-Wegner duality
S, which is the gauging operation underlying non-invertible duality symmetries, and the stacking of a 1-form symmetry SPT
T. In the continuum, they form a central extension of
PSL(2,Z4) for
p=2, and of
SL(2,Zp) for odd primes
p, whose central elements are invertible theories with purely gravitational response. These central extensions are governed by a twisted, graded generalization of the Witt group of abelian anyon theories, which we determine. For
p=2 the resulting group is the single-qubit Clifford group, with duality and entangler acting as the Hadamard and phase gates. We realize this entire structure microscopically as quantum cellular automata (QCA) acting on a certain local operator algebra associated with a spin lattice Hilbert space on a cubic lattice. Specifically, our local operator algebra is built by starting with all local operators commuting with a
Zp 1-form symmetry, and taking the quotient by all the (local) 1-form symmetry generators. The central elements can always be extended to the full tensor product algebra with a uniquely defined QCA class. For
p=2 they are generated by the non-trivial semion QCA, and for odd prime
p they are generated by the non-trivial
Zp Clifford QCA. Consequently the lattice fusion rules reproduce the continuum ones only up to these QCAs and lattice translations, giving rise to fusion rules refined by QCAs.
We have generalized the well-known statement that the Clifford group is a unitary 3-design into symmetric cases by extending the notion of unitary design.Concretely, we have proven that a symmetric Clifford group is a symmetric unitary 3-design if and only if the symmetry constraint is described by some Pauli subgroup.We have also found a complete and unique construction method of symmetric Clifford groups with simple quantum gates for Pauli symmetries.For the overall understanding, we have also considered physically relevant U(1) and SU(2) symmetry constraints, which cannot be described by a Pauli subgroup, and have proven that the symmetric Clifford group is a symmetric unitary 1-design but not a 2-design under those symmetries.Our findings are numerically verified by computing the frame potentials, which measure the difference in randomness between the uniform ensemble on the symmetric group of interest and the symmetric unitary group.This work will open a new perspective into quantum information processing, such as randomized benchmarking, and give a deep understanding to many-body systems, such as monitored random circuits.
The quantum theory of a finite quantum system with L degrees of freedom is usually set up by associating it with a Hilbert space of dimension d ( L ) and identifying observables and states in the matrix algebra . For the case d = 2 m , m integer, this algebra can be identified with the Clifford algebra . The case of d = 2 m dimensions is simply realized by a system with m dichotomic degrees of freedom, an m -qubit system for instance. The physically relevant new point is the appearance of a new (symmetry-?)group SO (2 m ). A possible interpretation of the space in which this group operates is proposed. It is shown that the eigenvalues of m -qubit-type states only depend on SO (2 m )-invariants. We use this fact to determine state parameter domains (generalized Bloch spheres) for states classified as SO (2 m )-tensors. The classification of states and interactions of components of a physical m -qubit system as k -tensors and pseudotensors (0 ⩽ k ⩽ m ) leads to rules similar to those found in elementary quantum mechanics. The question of electromagnetic interactions is shortly broached. We sketch, pars pro toto, a graphical interpretation of tensor contractions appearing in perturbative expansions.