Arthur Mehta

We show that it is undecidable to determine whether the commuting operator value of a nonlocal game is strictly greater than 1/2. As a corollary, there is a boolean constraint system (BCS) nonlocal game for which the value of the Navascués, Pironio, and Acín (NPA) hierarchy does not attain the commuting operator value at any finite level. Our contribution involves establishing a computable mapping from Turing machines to BCS nonlocal games in which the halting property of the machine is encoded as a decision problem for the commuting operator value of the game. Our techniques are algebraic and distinct from those used to establish MIP*=RE. As a first step, we construct a mapping from Turing machines to elements of the tensor product of free algebras, showing that deciding positivity of those elements is coRE-hard. As a second step, we extend this mapping to further realize these elements as game polynomials for BCS games.

Problems based on the structure of graphs – for example finding cliques, independent sets, or colourings – are of fundamental importance in classical complexity. It is well motivated to consider similar problems about quantum graphs, which are an operator system generalisation of graphs. Defining well-formulated decision problems for quantum graphs faces several technical challenges, and consequently the connections between quantum graphs and complexity have been underexplored. In this work, we introduce and study the clique problem for quantum graphs. Our approach utilizes a well-known connection between quantum graphs and quantum channels. The inputs for our problems are presented as quantum channels induced by circuits, which implicitly determine a corresponding quantum graph. We also use this approach to reimagine the clique and independent set problems for classical graphs, by taking the inputs to be circuits of deterministic or noisy channels which implicitly determine confusability graphs. We show that, by varying the collection of channels in the language, these give rise to complete problems for the classes NP, MA, QMA, and QMA(2). In this way, we exhibit a classical complexity problem whose natural quantisation is QMA(2), rather than QMA, which is commonly assumed. To prove the results in the quantum case, we make use of methods inspired by self-testing. To illustrate the utility of our techniques, we include a new proof of the reduction of QMA(k) to QMA(2) via cliques for quantum graphs. We also study the complexity of a version of the independent set problem for quantum graphs, and provide preliminary evidence that it may be in general weaker in complexity, contrasting to the classical case where the clique and independent set problems are equivalent.

Given that reliable cloud quantum computers are becoming closer to reality, the concept of delegation of quantum computations and its verifiability is of central interest. Many models have been proposed, each with specific strengths and weaknesses. Here, we put forth a new model where the client trusts only its classical processing, makes no computational assumptions, and interacts with a quantum server in a single round. In addition, during a set-up phase, the client specifies the size n of the computation and receives an untrusted, off-the-shelf (OTS) quantum device that is used to report the outcome of a single measurement. We show how to delegate polynomial-time quantum computations in the OTS model. This also yields an interactive proof system for all of QMA, which, furthermore, we show can be accomplished in statistical zero-knowledge. This provides the first relativistic (one-round), two-prover zero-knowledge proof system for QMA. As a proof approach, we provide a new self-test for n EPR pairs using only constant-sized Pauli measurements, and show how it provides a new avenue for the use of simulatable codes for local Hamiltonian verification. Along the way, we also provide an enhanced version of a well-known stability result due to Gowers and Hatami and show how it completes a common argument used in self-testing.

Nonlocal games are a foundational tool for understanding entanglement and constructing quantum protocols in settings with multiple spatially separated quantum devices. In this work, we continue the study initiated by Kalai et al. (STOC '23) of compiled nonlocal games, played between a classical verifier and a single cryptographically limited quantum device. Our main result is that the compiler proposed by Kalai et al. is sound for any two-player XOR game. A celebrated theorem of Tsirelson shows that for XOR games, the quantum value is exactly given by a semidefinite program, and we obtain our result by showing that the SDP upper bound holds for the compiled game up to a negligible error arising from the compilation. This answers a question raised by Natarajan and Zhang (FOCS '23), who showed soundness for the specific case of the CHSH game. Using our techniques, we obtain several additional results, including (1) tight bounds on the compiled value of parallel-repeated XOR games, (2) operator self-testing statements for any compiled XOR game, and (3) a ``nice" sum-of-squares certificate for any XOR game, from which operator rigidity is manifest.

Given that reliable cloud quantum computers are becoming closer to reality, the concept of delegation of quantum computations and its verifiability is of central interest. Many models have been proposed, each with specific strengths and weaknesses. Here, we put forth a new model where the client trusts only its classical processing, makes no computational assumptions, and interacts with a quantum server in a \emph{single} round. In addition, during a set-up phase, the client specifies the size $n$ of the computation and receives an untrusted, \emph{off-the-shelf (OTS)} quantum device that is used to report the outcome of a single constant-sized measurement from a predetermined logarithmic-sized input. In the OTS model, we thus picture that a single quantum server does the bulk of the computations, while the OTS device is used as an untrusted and generic verification device, all in a single round. We show how to delegate polynomial-time quantum computations in the OTS model. Scaling up the technique also yields an interactive proof system for all of QMA, which, furthermore, we show can be accomplished in statistical zero-knowledge. This yields the first relativistic (one-round), two-prover zero-knowledge proof system for QMA. As a proof approach, we provide a new self-test for $n$-EPR pairs using only constant-sized Pauli measurements, and show how it provides a new avenue for the use of simulatable codes for local Hamiltonian verification. Along the way, we also provide an enhanced version of a well-known stability result due to Gowers and Hatami and show how it completes a common argument used in self-testing.

In a proof of knowledge (PoK), a verifier becomes convinced that a prover possesses privileged information. In combination with zero- knowledge proof systems, PoKs are an important part of secure protocols such as digital signature schemes and authentication schemes as they en- able a prover to demonstrate posession of a certain piece of information (such as a private key or a credential), without revealing it. Formally, A PoK is defined via the existence of an extractor, which is capable of recon- structing the key information that makes a verifier accept, given oracle access to the prover. We extend the concept of a PoK in the setting of a single classical verifier and two quantum provers, and exhibit the PoK property for the Hamil- tonian game, a non-local game between a single classical verifier and two quantum provers for the local Hamiltonian problem. More specifically, we construct an extractor which, given oracle access to a provers’ strategy that leads to high acceptance probability, is able to reconstruct the ground state of a local Hamiltonian. Our result can be seen as a new form of self- testing, where, in addition to certifying a pre-shared entangled state and the prover’s strategy, the verifier also certifies a local quantum state. This technique thus provides a method to ascertain that a prover has access to a quantum system, in particular, a ground state, Thus indicating a new level of verification for a proof of quantumness.