quantum protocol designers

Abstract: We confirm a conjecture of the first author [SODA'26] on a strong form of pseudo-telepathy for Khot's $d$-to-$1$ games: for every $d\geq 4$, every $ε>0$, and every $k\in\mathbb N$, there is a $d$-to-$1$ label-cover game of classical value less than $ε$ that admits a perfect quantum strategy using finite-dimensional shared entanglement with measurement contexts of radius at least $k$. As a consequence, there exist graphs whose quantum chromatic number $χ_q$ is $3$ and whose classical chromatic number $χ$ is arbitrarily large. This is the first improvement on the quantum chromatic gap since the work of Brassard, Cleve, and Tapp [Phys. Rev. Lett.'99], which produced graphs for which $χ$ grows exponentially in $χ_q$. To prove the conjecture, we proceed as follows. We start with a ``stretched'' version of the Mermin--Peres magic square, obtained by spacing out its equations and linking them with additional long equation wires. This operation increases the radius of the initial measurement contexts of a perfect quantum strategy for the game, while preserving classical unsatisfiability. We then apply parallel repetition to make the classical value small, and we turn the system into a pseudo-telepathic multi-layered Kneser hypergraph via a PCP construction of Dinur, Guruswami, Khot, and Regev [STOC'03]. Finally, we feed the resulting hypergraph into the reduction of Fei, Minzer, and Wang from hypergraph colouring to $4$-to-$1$ games, we establish quantum completeness of this reduction, and we show that it preserves context radius.

Abstract: We study the one-way communication complexity of the $f$-Boolean Hidden Partition problem. Here, Alice is given an $n$-bit string and Bob is given $Ω(n)$ disjoint blocks of its indices together with a string of labels. Under the promise that the evaluations of $f$ on these blocks either agree with all of the labels or disagree with all of them, Bob must determine which is the case using a single message from Alice. This problem generalizes the Boolean Hidden Matching and Hidden Hypermatching problems, which capture the special case where $f$ is the parity function. We establish classical and quantum communication bounds for $f$-Boolean Hidden Partition in terms of the sign degree $d$ of $f$, proving a conjecture of Doriguello and Montanaro (TQC 2020). Their prior work gave logarithmic communication upper bounds for classical protocols when $d \le 1$ and quantum protocols when $d \le 2$, as well as polynomial lower bounds for certain structured functions with larger sign degree. We show that for every $f$ of sign degree $d \ge 2$, the randomized classical communication complexity of this problem is $Θ(n^{1-1/d})$ while its quantum communication complexity lies between $Ω(n^{1-2/d})$ and $\bO{n^{1-1/\lceil d/2 \rceil} \log n}$. This completely characterizes the functions $f$ admitting polynomial quantum advantage as those for which $d \ge 2$, with a new infinite family of exponential separations given by the case $d = 2$.

Abstract: We study interactive multiprover games and many-round protocols in which the communication between the verifier and the provers is quantum. Parallel repetition is known to suppress soundness error of classical two-prover games arbitrarily close to zero, as shown by Raz (STOC '95), even with quantum strategies as shown by Yuen (ICALP '16), and Bavarian, Vidick and Yuen (STOC '17). Yet, we show that quantum communication can have the opposite, unexpected effect. Specifically, we exhibit a one-round quantum game and two-round $\mathsf{QMIP}$ protocols whose local (unentangled) value is strictly less than one for a single instance, yet equals one under $n$-fold parallel repetition for every $n \geq 2$. The key is that parallel repetition not only imposes additional winning conditions but also supplies additional quantum resources as entanglement in the exchanged states can be exploited jointly across copies. Our multiround examples build on superactivation of zero-error capacities of quantum channels. To establish this connection, we introduce quantum games and interactive protocols which capture the one-shot zero-error classical and quantum capacities, both with and without entanglement assistance. Our constructions use quantum-state verification and a teleportation-based reduction from two rounds to one. In contrast, when shared entanglement is allowed, we show that parallel repetition cannot increase the entangled value when at most $3$ messages are exchanged, consistent with classical results of Bellare, Impagliazzo and Naor (FOCS '97), and that of single-prover systems by Kitaev and Watrous (STOC '00). Combining this monotonicity with our quantum game-channel correspondence, we show that entanglement-assisted zero-error classical and quantum capacities cannot be superactivated.

Abstract: We prove that the quantum monad in dimension $2n$ admits no natural transformation to the polymorphism clone of linear equations modulo $n$. Consequently, for every $n\geq 2$, there exists an unsatisfiable system of linear equations over $\mathbb{Z}_n$ whose constraint system game admits a perfect finite-dimensional quantum strategy. As a corollary, we completely characterise pseudo-telepathic constraint languages in finite dimension. The proof combines a result of Harding, Jager, and Smith on group-valued measures on subspaces of Hilbert spaces with the polymorphism-minion characterisation of quantum pseudo-telepathy.