A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang–Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.
A new kind of quantum gates, higher braiding gates, are introduced and studied. These are matrix solutions of the polyadic braid equations. Such gates support a special kind of multi-qubit entanglement which can speed up key distribution and accelerate the execution of algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates which can be related to qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). The higher braid group and its representation by higher braid operators are given.
Commutativity gadgets provide a technique for lifting classical reductions between constraint satisfaction problems to quantum-sound reductions between the corresponding nonlocal games. We develop a general framework for commutativity gadgets in the setting of quantum homomorphisms between finite relational structures. Building on the notion of quantum homomorphism spaces, we introduce a uniform notion of commutativity gadget capturing the finite-dimensional quantum, quantum approximate, and commuting-operator models. In the robust setting, we use the weighted-algebra formalism for approximate quantum homomorphisms to capture corresponding notions of robust commutativity gadgets.
Our main results characterize both non-robust and robust commutativity gadgets purely in terms of quantum polymorphism spaces: in any model, existence of a commutativity gadget is equivalent to the collapse of the corresponding quantum polymorphisms to classical ones at arity
∣A∣2, and robust gadgets are characterized by stable commutativity of the appropriate weighted polymorphism algebra. We use this characterisation to show relations between the classes of commutativity gadget, notably that existence of a robust commutativity gadget is equivalent to the existence of a corresponding non-robust one.
Finally, we prove that quantum polymorphisms of complete graphs
Kn have a very special structure, wherein the noncommutative behaviour only comes from the quantum permutation group
Sn+. Combining this with techniques from combinatorial group theory, we construct separations between commutativity-gadget classes: we exhibit a relational structure admitting a finite-dimensional commutativity gadget but no quantum approximate gadget, and, conditional on the existence of a non-hyperlinear group, a structure admitting a quantum approximate commutativity gadget but no commuting-operator gadget.
Recently, Mančinska and Roberson proved that two graphs G and
G′ are quantum isomorphic if and only if they admit the same number of homomorphisms from all planar graphs. We extend this result to planar #CSP with any pair of sets
F and
F′ of real-valued, arbitrary-arity constraint functions. Graph homomorphism is the special case, in which each of
F and
F′ contains a single symmetric 0-1-valued binary constraint function. Our treatment uses the framework of planar Holant problems. To prove that quantum isomorphic constraint function sets give the same value on any planar #CSP instance, we apply a novel form of holographic transformation of Valiant, using the quantum permutation matrix
U defining the quantum isomorphism. Due to the noncommutativity of
U ’s entries, it turns out that this form of holographic transformation is only applicable to planar Holant. To prove the converse, we introduce the quantum automorphism group
Qut(F) of a set
F of constraint functions/tensors and characterize the intertwiners of
Qut(F) as the signature matrices of planar \(\text{Holant}{\mathcal {F}\,|\,\mathcal {EQ})\) quantum gadgets. Then, we define a new notion of (projective) connectivity for constraint functions and reduce their arity while preserving their inclusion in the original intertwiner space. Finally, to address the challenges posed by generalizing from 0-1 valued to real-valued constraint functions, we adapt a technique of Lovász in the classical setting for isomorphisms of real-weighted graphs to the setting of quantum isomorphisms.
Let x○y and x⊕y denote complex valued generalized multiplication and addition of complex numbers. If (a) multiplication is associative, (b) addition is associative and commutative, (c) there exists a complex number e satisfying x⊕e=x for all x, (d) multiplication is distributive with respect to addition, and (e) both operations are analytic functions in the two variables, then there exists a mapping x′→x such that (x○y)′=x′⋅y′ and (x⊕y)′=x′+y′. This result is used in the derivation of the multiplication and addition laws for quantum amplitudes.