In complexity theory, gap-preserving reductions play a crucial role in studying hardness of approximation and in analyzing the relative complexity of multiprover interactive proof systems. In the quantum setting, multiprover interactive proof systems with entangled provers correspond to gapped promise problems for nonlocal games, and the recent result MIP*=RE [1] shows that these are in general undecidable. However, the relative complexity of problems within MIP* is still not well-understood, as establishing gap-preserving reductions in the quantum setting presents new challenges. In this paper, we introduce a framework to study such reductions and use it to establish MIP*-completeness of the gapped promise problem for the natural class of independent set games. In such a game, the goal is to determine whether a given graph contains an independent set of a specified size. We construct families of independent set games with constant question size for which the gapped promise problem is undecidable. In contrast, the same problem is decidable in polynomial time in the classical setting. To carry out our reduction, we establish a new stability theorem, which could be of independent interest, allowing us to perturb families of aThis is a striking phenomenlmost PVMs to genuine PVMs.

Majority vote is a basic method for amplifying correct outcomes that is widely used in computer science and beyond. While it can amplify the correctness of a quantum device with classical output, the analogous procedure for quantum output is not known. We introduce quantum majority vote as the following task: given a product state |ψ_1⟩ ⊗ … ⊗ |ψ_n⟩ where each qubit is in one of two orthogonal states |ψ⟩ or |ψ^⟂⟩, output the majority state. We show that an optimal algorithm for this problem achieves worst-case fidelity of 1/2 + Θ(1/√n). Under the promise that at least 2/3 of the input qubits are in the majority state, the fidelity increases to 1 - Θ(1/n) and approaches 1 as n increases.We also consider the more general problem of computing any symmetric and equivariant Boolean function f: {0,1}ⁿ → {0,1} in an unknown quantum basis, and show that a generalization of our quantum majority vote algorithm is optimal for this task. The optimal parameters for the generalized algorithm and its worst-case fidelity can be determined by a simple linear program of size O(n). The time complexity of the algorithm is O(n⁴ log n) where n is the number of input qubits.

Over 50 years ago, Lovász proved that two graphs are isomorphic if and only if they admit the same number of homomorphisms from any graph. Other equivalence relations on graphs, such as cospectrality or fractional isomorphism, can be characterized by equality of homomorphism counts from an appropriately chosen class of graphs. Dvořák [J. Graph Theory 2010] showed that taking this class to be the graphs of treewidth at most k yields a tractable relaxation of graph isomorphism known as k-dimensional Weisfeiler-Leman equivalence. Together with a famous result of Cai, Fürer, and Immerman [FOCS 1989], this shows that homomorphism counts from graphs of bounded treewidth do not determine a graph up to isomorphism. Dell, Grohe, and Rattan [ICALP 2018] raised the questions of whether homomorphism counts from planar graphs determine a graph up to isomorphism, and what is the complexity of the resulting relation. We answer the former in the negative by showing that the resulting relation is equivalent to the so-called quantum isomorphism [Mančinska et al, ICALP 2017]. Using this equivalence, we further resolve the latter question, showing that testing whether two graphs have the same number of homomorphisms from any planar graph is, surprisingly, an undecidable problem, and moreover is complete for the class coRE (the complement of recursively enumerable problems). Quantum isomorphism is defined in terms of a one-round, two-prover interactive proof system in which quantum provers, who are allowed to share entanglement, attempt to convince the verifier that the graphs are isomorphic. Our combinatorial proof leverages the quantum automorphism group of a graph, a notion from noncommutative mathematics.

We introduce a nonlocal game that captures and extends the notion of graph isomorphism. This game can be won in the classical case if and only if the two input graphs are isomorphic. Thus, by considering quantum strategies we are able to define the notion of quantum isomorphism. We also consider the case of more general non-signalling strategies, and show that such a strategy exists if and only if the graphs are fractionally isomorphic. We prove several necessary conditions for quantum isomorphism, including cospectrality, and provide a construction for producing pairs of non-isomorphic graphs that are quantum isomorphic. We then show that both classical and quantum isomorphism can be reformulated as feasibility programs over the completely positive and completely positive semidefinite cones respectively. This leads us to considering relaxations of (quantum) isomorphism arrived at by relaxing the cone to either the doubly nonnegative (DNN) or positive semidefinite (PSD) cones. We show that DNN-isomorphism is equivalent to the previous defined notion of graph equivalence, a polynomial-time decidable relation that is related to coherent algebras. We also show that PSD-isomorphism implies several types of cospectrality, and that it is equivalent to cospectrality for connected 1-walk-regular graphs. Finally, we show that all of the above mentioned relations form a strict hierarchy of weaker and weaker relations, with non-singalling/fractional isomorphism being the weakest. The techniques used are an interesting mix of algebra, combinatorics, and quantum information.

We classify two-qubit commuting Hamiltonians in terms of their computational complexity. Suppose one has a two-qubit commuting Hamiltonian H which one can apply to any pair of qubits, starting in a computational basis state. We prove a dichotomy theorem: either this model is efficiently classically simulable or it allows one to sample from probability distributions which cannot be sampled from classically unless the polynomial hierarchy collapses. Furthermore, the only simulable Hamiltonians are those which fail to generate entanglement. This shows that generic two-qubit commuting Hamiltonians can be used to perform computational tasks which are intractable for classical computers under plausible assumptions. Our proof makes use of new postselection gadgets and Lie theory.

We study zero-error entanglement-assisted source-channel coding (communication in the presence of side information). Adapting a technique of Beigi, we show that such coding requires existence of a set of vectors satisfying orthogonality conditions related to suitably defined graphs G and H. Such vectors exist if and only if ϑ(G̅) ≤ ϑ(H̅), where ϑ represents the Lovász number. We also obtain similar inequalities for the related Schrijver ϑ-and Szegedy ϑ+numbers. These inequalities reproduce several known bounds and also lead to new results. We provide a lower bound on the entanglement-assisted cost rate. We show that the entanglement-assisted independence number is bounded by the Schrijver number: α*(G) ≤ ϑ-(G). Therefore, we are able to disprove the conjecture that the one-shot entanglement-assisted zero-error capacity is equal to the integer part of the Lovász number. Beigi introduced a quantity β as an upper bound on α* and posed the question of whether β(G) = ⌊ϑ(G)⌋. We answer this in the affirmative and show that a related quantity is equal to ⌊ϑ(G)⌋. We show that a quantity χvect(G) recently introduced in the context of Tsirelson's problem is equal to ⌊ϑ+(G)⌋. In an appendix, we investigate multiplicativity properties of Schrijver's and Szegedy's numbers, as well as projective rank.

We introduce two generalizations of Kochen-Specker (KS) sets: projective KS sets and generalized KS sets. We then use projective KS sets to characterize all graphs for which the chromatic number is strictly larger than the quantum chromatic number. Here, the quantum chromatic number is defined via a nonlocal game based on graph coloring. We further show that from any graph with separation between these two quantities, one can construct a classical channel for which entanglement assistance increases the one-shot zero-error capacity. As an example, we exhibit a new family of classical channels with an exponential increase.