arXiv subjects

Kieran Mastel

Publications and source records attributed to Kieran Mastel.

7 recordsLinked to original sources

arXiv2025

The Clifford theory of the n-qubit Clifford group

The n-qubit Pauli group and its normalizer the n-qubit Clifford group have applications in quantum error correction and device characterization. Recent applications have made use of the representation theory of the Clifford group. We apply the tools of (the coincidentally named) Clifford theory to examine the representation theory of the Clifford group using the much simpler representation theory of the Pauli group. We find an unexpected correspondence between irreducible characters of the n-qubit Clifford group and those of the (n+1)-qubit Clifford group.

quant-ph↗

arXiv2025

The quantum smooth label cover problem is undecidable

We show that the quantum smooth label cover problem is undecidable and RE-hard. This sharply contrasts the quantum unique label cover problem, which can be decided efficiently by a result of Kempe, Regev, and Toner (FOCS'08). On the other hand, our result aligns with the RE-hardness of the quantum label cover problem, which follows from the celebrated MIP* = RE result of Ji, Natarajan, Vidick, Wright, and Yuen (ACM'21). Additionally, we show that the quantum oracularized smooth label cover problem is RE-hard. Our second result fits with the alternative quantum unique games conjecture recently proposed by Mousavi and Spirig (ITCS'25) on the RE-hardness of the quantum oracularized unique label cover problem. Our proof techniques include a quantum version of Feige's reduction from 3SAT to 3SAT5 (STOC'96) for BCSMIP*-protocols, which may be of independent interest.

quant-ph↗

arXiv2025

Succinct Perfect Zero-knowledge for MIP*

In their recent breakthrough result, Slofstra and the second author show that there is a two-player one-round perfect zero-knowledge MIP* protocol for RE (STOC'24). We build on their result to show that there exists a succinct two-player one-round perfect zero-knowledge MIP* protocol for RE against dishonest verifiers with polylog question size and O(1) answer size, or with O(1) question size and polylog answer size. To prove our result, we study the three central compression techniques underlying the MIP*=RE proof (Ji et al. '20): question reduction, oracularization, and answer reduction. We show that question reduction preserves the perfect (as well as statistical and computational) zero-knowledge properties of the original protocol against dishonest verifiers, and oracularization and answer reduction preserve the perfect (as well as statistical and computational) zero-knowledge properties of the original protocol against honest verifiers. Secondly, we show that every constraint-constraint binary constraint system (BCS) nonlocal game, which provides a quantum information characterization of MIP*, can be converted to a synchronous constraint-variable BCS game to preserve perfect completeness for our compression. Lastly, we present a parametrized perfect-zero-knowledge transformation of MIP* protocols, which generalizes the transformation in (Slofstra and Kieran STOC'24) . This transformation allows us to preserve the zero-knowledge property against dishonest verifiers in the recursively oracularized protocols in our compression.

quant-ph↗

arXiv2025

RE-completeness of entangled constraint satisfaction problems

Constraint satisfaction problems (CSPs) are a natural class of decision problems where one must decide whether there is an assignment to variables that satisfies a given formula. Schaefer's dichotomy theorem, and its extension to all alphabets due to Bulatov and Zhuk, shows that CSP languages are either efficiently decidable, or NP-complete. It is possible to extend CSP languages to quantum assignments using the formalism of nonlocal games. Due to the equality of complexity classes MIP$^\ast=$ RE, general succinctly-presented entangled CSPs are RE-complete. In this work, we show that a wide range of NP-complete CSPs become RE-complete in this setting, including all boolean CSPs, such as 3SAT, as well as $3$-colouring. This also implies that these CSP languages remain undecidable even when not succinctly presented. To show this, we work in the weighted algebra framework introduced by Mastel and Slofstra, where synchronous strategies for a nonlocal game are represented by tracial states on an algebra. Along the way, we improve the subdivision technique in order to be able to separate constraints in the CSP while preserving constant soundness, construct commutativity gadgets for all boolean CSPs, and show a variety of relations between the different ways of presenting CSPs as games.

quant-ph↗

arXiv2024

Two prover perfect zero knowledge for MIP*

The recent MIP*=RE theorem of Ji, Natarajan, Vidick, Wright, and Yuen shows that the complexity class MIP* of multiprover proof systems with entangled provers contains all recursively enumerable languages. Prior work of Grilo, Slofstra, and Yuen [FOCS '19] further shows (via a technique called simulatable codes) that every language in MIP* has a perfect zero knowledge (PZK) MIP* protocol. The MIP*=RE theorem uses two-prover one-round proof systems, and hence such systems are complete for MIP*. However, the construction in Grilo, Slofstra, and Yuen uses six provers, and there is no obvious way to get perfect zero knowledge with two provers via simulatable codes. This leads to a natural question: are there two-prover PZK-MIP* protocols for all of MIP*? In this paper, we show that every language in MIP* has a two-prover one-round PZK-MIP* protocol, answering the question in the affirmative. For the proof, we use a new method based on a key consequence of the MIP*=RE theorem, which is that every MIP* protocol can be turned into a family of boolean constraint system (BCS) nonlocal games. This makes it possible to work with MIP* protocols as boolean constraint systems, and in particular allows us to use a variant of a construction due to Dwork, Feige, Kilian, Naor, and Safra [Crypto '92] which gives a classical MIP protocol for 3SAT with perfect zero knowledge. To show quantum soundness of this classical construction, we develop a toolkit for analyzing quantum soundness of reductions between BCS games, which we expect to be useful more broadly. This toolkit also applies to commuting operator strategies, and our argument shows that every language with a commuting operator BCS protocol has a two prover PZK commuting operator protocol.

quant-ph↗

arXiv2023

A New Class of Higher Quantum Airy Structures as Modules of $\mathcal{W}(\mathfrak{gl}_r)$-Algebras

Quantum $r$-Airy structures can be constructed as modules of $\mathcal{W}(\mathfrak{gl}_r)$-algebras via restriction of twisted modules for the underlying Heisenberg algebra. In this paper we classify all such higher quantum Airy structures that arise from modules twisted by automorphisms of the Cartan subalgebra consisting of products of disjoint cycles of the same length. An interesting feature of these higher quantum Airy structures is that the dilaton shifts must be chosen carefully to satisfy a matrix invertibility condition, with a natural choice being roots of unity. We explore how these higher quantum Airy structures may provide a definition of the Chekhov, Eynard, and Orantin topological recursion for reducible algebraic spectral curves. We also study under which conditions quantum $r$-Airy structures that come from modules twisted by arbitrary automorphisms can be extended to new quantum $(r+1)$-Airy structures by appending a trivial one-cycle to the twist without changing the dilaton shifts.

math-ph↗

arXiv2019

Deep Learning-Guided Surface Characterization for Autonomous Hydrogen Lithography

As the development of atom scale devices transitions from novel, proof-of-concept demonstrations to state-of-the-art commercial applications, automated assembly of such devices must be implemented. Here we present an automation method for the identification of defects prior to atomic fabrication via hydrogen lithography using deep learning. We trained a convolutional neural network to locate and differentiate between surface features of the technologically relevant hydrogen-terminated silicon surface imaged using a scanning tunneling microscope. Once the positions and types of surface features are determined, the predefined atomic structures are patterned in a defect-free area. By training the network to differentiate between common defects we are able to avoid charged defects as well as edges of the patterning terraces. Augmentation with previously developed autonomous tip shaping and patterning modules allows for atomic scale lithography with minimal user intervention.

cond-mat.mtrl-sci↗