More Gadgets from the Quantum Spookhouse1

Eric Culf, Josse van Dobben de Bruyn, Peter Zeman

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 $K_n$ have a very special structure, wherein the noncommutative behaviour only comes from the quantum permutation group $S_n^+$. 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.

Peter Weiß

Shamini Bundell

Eric Culf, Josse van Dobben de Bruyn, Matthijs Vernooij +1 more

Commutativity gadgets allow NP-hardness proofs for classical constraint satisfaction problems (CSPs) to be carried over to undecidability proofs for the corresponding entangled CSPs. This has been done, for instance, for NP-complete boolean CSPs and 3-colouring in the work of Culf and Mastel. For many CSPs over larger alphabets, including $k$-colouring when $k \geq 4$, it is not known whether or not commutativity gadgets exist, or if the entangled CSP is decidable. In this paper, we initiate a systematic study of commutativity gadgets and prove the first known obstruction to their existence. We do this by extending the definition of the quantum automorphism group of a graph to the quantum endomorphism monoid of a CSP, and showing that a CSP with non-classical quantum endomorphism monoid does not admit a commutativity gadget. As a consequence of this, we get that no commutativity gadget exists for $k$-colouring when $k \geq 4$. However, we construct a commutativity gadget for an alternate way of presenting $k$-colouring as a nonlocal game, the oracular setting. Furthermore, we prove an easy to check sufficient condition for the quantum endomorphism monoid to be non-classical, extending a result of Schmidt for the quantum automorphism group of a graph, and use this to give examples of CSPs that do not admit a commutativity gadget. We also show that existence of commutativity gadgets and oracular commutativity gadgets is equivalent for graphs with no four-cycle; and that the odd cycles and the odd graphs have a commutative quantum endomorphism monoid, leaving open the possibility that they might admit a commutativity gadget.

Banks, Robert J., Natasha Feinstein, Roopayan Ghosh +2 more

In certain scenarios, quantum annealing can be made more efficient by additional $XX$ interactions. It has been shown that the additional interactions can reduce the scaling of perturbative crossings. In traditional annealing devices these couplings do not exist natively. In this work, we develop two gadgets to achieve this: a three-body gadget that requires a strong $ZZZ$ interaction; and a one-hot gadget that uses only local $X$ drives and two-body $ZZ$ interactions. The gadgets partition the Hilbert space to effectively generate a limited number of $XX$ interactions in the low-energy subspace. We numerically verify that the one-hot gadget can mitigate a perturbative crossing on a toy problem. These gadgets establish new pathways for implementing and exploiting $XX$ interactions, enabling faster and more robust quantum annealing.