Alex Grilo | LIP6 - Équipe QI

Complexity of geometrically local stoquastic Hamiltonians

Wed, 11 Feb 2026 00:00:00 +0000

The QMA-completeness of the local Hamiltonian problem is a landmark result of the field of Hamiltonian complexity that studies the computational complexity of problems in quantum many-body physics. Since its proposal, substantial effort has been invested in better understanding the problem for physically motivated important families of Hamiltonians. In particular, the QMA-completeness of approximating the ground state energy of local Hamiltonians has been extended to the case where the Hamiltonians are geometrically local in one and two spatial dimensions. Among those physically motivated Hamiltonians, stoquastic Hamiltonians play a particularly crucial role, as they constitute the manifestly sign-free Hamiltonians in Monte Carlo approaches. Interestingly, for such Hamiltonians, the problem at hand becomes more ‘‘classical’’, being hard for the class MA (the randomized version of NP) and its complexity has tight connections with derandomization. In this work, we prove that both the two- and one-dimensional geometrically local analogues remain MA-hard with high enough qudit dimension. Moreover, we show that related problems are StoqMA-complete.

Computational Monogamy of Entanglement and Non-interactive Quantum Key Distribution

Mon, 01 Dec 2025 00:00:00 +0000

Quantum key distribution (QKD) enables Alice and Bob to exchange a secret key over a public, untrusted quantum channel. Compared to classical key exchange, QKD achieves everlasting security: after the protocol execution the key is secure against adversaries that can do unbounded computations. On the flip side, while classical key exchange can be achieved non-interactively (with two simultaneous messages between Alice and Bob), no non-interactive protocol is known that provides everlasting security, even using quantum information.

Post-Quantum Zero-Knowledge with Space-Bounded Simulation

Wed, 01 Oct 2025 00:00:00 +0000

The traditional definition of quantum zero-knowledge stipulates that the knowledge gained by any quantum polynomial-time verifier in an interactive protocol can be simulated by a quantum polynomial-time algorithm. One drawback of this definition is that it allows the simulator to consume significantly more computational resources than the verifier. We argue that this drawback renders the existing notion of quantum zero-knowledge not viable for certain settings, especially when dealing with near-term quantum devices.

The Role of Piracy in Quantum Proofs

Wed, 01 Oct 2025 00:00:00 +0000

A well-known feature of quantum information is that it cannot, in general, be cloned. Recently, a number of quantum-enabled information-processing tasks have demonstrated various forms of uncloneability; among these forms, piracy is an adversarial model that gives maximal power to the adversary in controlling both a cloning-type attack, as well as the evaluation/verification stage. Here, we initiate the study of anti-piracy proof systems, which are proof systems that inherently prevent piracy attacks. We define anti-piracy proof systems, demonstrate such a proof system for an oracle problem, and also describe a candidate anti-piracy proof system for {$}{$}{\backslash}textsf {{}NP {}} {$}{$}NP. We also study quantum proof systems that are cloneable and settle the famous QMA vs. {$}{$}{\backslash}textsf {{}QMA {}} (2){$}{$}QMA(2)debate in this setting. Lastly, we discuss how one can approach the QMA vs. QCMA question, by studying its cloneable variants.

Trainability and Expressivity of Hamming-Weight Preserving Quantum Circuits for Machine Learning

Thu, 15 May 2025 00:00:00 +0000

Quantum machine learning (QML) has become a promising area for real world applications of quantum computers, but near-term methods and their scalability are still important research topics. In this context, we analyze the trainability and controllability of specific Hamming weight preserving variational quantum circuits (VQCs). These circuits use qubit gates that preserve subspaces of the Hilbert space, spanned by basis states with fixed Hamming weight k . In this work, we first design and prove the feasibility of new heuristic data loaders, performing quantum amplitude encoding of ( n k ) -dimensional vectors by training an n -qubit quantum circuit. These data loaders are obtained using controllability arguments, by checking the Quantum Fisher Information Matrix (QFIM)’s rank. Second, we provide a theoretical justification for the fact that the rank of the QFIM of any VQC state is almost-everywhere constant, which is of separate interest. Lastly, we analyze the trainability of Hamming weight preserving circuits, and show that the variance of the l 2 cost function gradient is bounded according to the dimension ( n k ) of the subspace. This proves conditions of existence/lack of Barren Plateaus for these circuits, and highlights a setting where a recent conjecture on the link between controllability and trainability of variational quantum circuits does not apply.

The Round Complexity of Proofs in the Bounded Quantum Storage Model

Tue, 08 Apr 2025 00:00:00 +0000

The round complexity of interactive proof systems is a key question of practical and theoretical relevance in complexity theory and cryptography. Moreover, results such as QIP = QIP(3) (STOC'00) show that quantum resources significantly help in such a task. In this work, we initiate the study of round compression of protocols in the bounded quantum storage model (BQSM). In this model, the malicious parties have a bounded quantum memory and they cannot store the all the qubits that are transmitted in the protocol. Our main results in this setting are the following: 1. There is a non-interactive (statistical) witness indistinguishable proof for any language in NP (and even QMA) in BQSM in the plain model. We notice that in this protocol, only the memory of the verifier is bounded. 2. Any classical proof system can be compressed in a two-message quantum proof system in BQSM. Moreover, if the original proof system is zero-knowledge, the quantum protocol is zero-knowledge too. In this result, we assume that the prover has bounded memory. Finally, we give evidence towards the “tightness” of our results. First, we show that NIZK in the plain model against BQS adversaries is unlikely with standard techniques. Second, we prove that without the BQS model there is no 2–message zero-knowledge quantum interactive proof, even under computational assumptions.

Subspace preserving quantum convolutional neural network architectures

Wed, 01 Jan 2025 00:00:00 +0000

Subspace preserving quantum circuits are a class of quantum algorithms that, relying on some symmetries in the computation, can offer theoretical guarantees for their training. Those algorithms have gained extensive interest as they can offer polynomial speed-up and can be used to mimic classical machine learning algorithms. In this work, we propose a novel convolutional neural network architecture model based on Hamming weight preserving quantum circuits. In particular, we introduce convolutional layers, and measurement based pooling layers that preserve the symmetries of the quantum states while realizing non-linearity using gates that are not subspace preserving. Our proposal offers significant polynomial running time advantages over classical deep-learning architecture. We provide an open source simulation library for Hamming weight preserving quantum circuits that can simulate our techniques more efficiently with GPU-oriented libraries. Using this code, we provide examples of architectures that highlight great performances on complex image classification tasks with a limited number of qubits, and with fewer parameters than classical deep-learning architectures.

Verifier-on-a-Leash: New Schemes for Verifiable Delegated Quantum Computation, with Quasilinear Resources

Mon, 01 Jan 2024 00:00:00 +0000

On-State Commutativity of Measurements and Joint Distributions of Their Outcomes

Wed, 27 Jan 2021 00:00:00 +0000

In this note, we analyze joint probability distributions that come from the outcomes of quantum measurements performed on sets of quantum states. First, we identify the properties of these distributions that need to be fulfilled to recover a classical behavior. Secondly, we connect the existence of a joint distribution with the “on-state” permutability (commutativity of more than two operators) of measurement operators. By “on-state” we mean properties of operators that can hold only on a subset of states in the Hilbert space. Then, we disprove a conjecture proposed by Carstens, Ebrahimi, Tabia, and Unruh (eprint 2018), which states that partial on-state permutation imply full on-state permutation. We disprove such a conjecture with a counterexample where pairwise “on-state” commutativity does not imply on-state permutability, unlike in the case where the definition is valid for all states in the Hilbert space. Finally, we explore the new concept of on-state commutativity by showing a simple proof that if two projections almost on-state commute, then there is a commuting pair of operators that are on-state close to the originals. This result was originally proven by Hasting (Communications in Mathematical Physics, 2019) for general operators.

Two combinatorial MA-complete problems

Wed, 06 Jan 2021 00:00:00 +0000

Despite the interest in the complexity class MA, the randomized analog of NP, just a few natural MA-complete problems are known. The first problem was found by (Bravyi and Terhal, SIAM Journal of Computing 2009); it was then followed by (Crosson, Bacon and Brown, PRE 2010) and (Bravyi, Quantum Information and Computation 2015). Surprisingly, two of these problems are defined using terminology from quantum computation, while the third is inspired by quantum computation and keeps a physical terminology. This prevents classical complexity theorists from studying these problems, delaying potential progress, e.g., on the NP vs. MA question. Here, we define two new combinatorial problems and prove their MA-completeness. The first problem, ACAC, gets as input a succinctly described graph, with some marked vertices. The problem is to decide whether there is a connected component with only unmarked vertices, or the graph is far from having this property. The second problem, SetCSP, generalizes standard constraint satisfaction problem (CSP) into constraints involving sets of strings. Technically, our proof that SetCSP is MA-complete is based on an observation by (Aharonov and Grilo, FOCS 2019), in which it was noted that a restricted case of Bravyi and Terhal’s problem (namely, the uniform case) is already MA-complete; a simple trick allows to state this restricted case using combinatorial language. The fact that the first, more natural, problem of ACAC is MA-hard follows quite naturally from this proof, while the containment of ACAC in MA is based on the theory of random walks. We notice that the main result of Aharonov and Grilo carries over to the SetCSP problem in a straightforward way, implying that finding a gap-amplification procedure for SetCSP (as in Dinur’s PCP proof) is equivalent to MA=NP. This provides an alternative new path towards the major problem of derandomizing MA.

Quantum statistical query learning

Mon, 07 Dec 2020 00:00:00 +0000

We propose a learning model called the quantum statistical learning QSQ model, which extends the SQ learning model introduced by Kearns to the quantum setting. Our model can be also seen as a restriction of the quantum PAC learning model: here, the learner does not have direct access to quantum examples, but can only obtain estimates of measurement statistics on them. Theoretically, this model provides a simple yet expressive setting to explore the power of quantum examples in machine learning. From a practical perspective, since simpler operations are required, learning algorithms in the QSQ model are more feasible for implementation on near-term quantum devices. We prove a number of results about the QSQ learning model. We first show that parity functions, (log n)-juntas and polynomial-sized DNF formulas are efficiently learnable in the QSQ model, in contrast to the classical setting where these problems are provably hard. This implies that many of the advantages of quantum PAC learning can be realized even in the more restricted quantum SQ learning model. It is well-known that weak statistical query dimension, denoted by WSQDIM(C), characterizes the complexity of learning a concept class C in the classical SQ model. We show that log(WSQDIM(C)) is a lower bound on the complexity of QSQ learning, and furthermore it is tight for certain concept classes C. Additionally, we show that this quantity provides strong lower bounds for the small-bias quantum communication model under product distributions. Finally, we introduce the notion of private quantum PAC learning, in which a quantum PAC learner is required to be differentially private. We show that learnability in the QSQ model implies learnability in the quantum private PAC model. Additionally, we show that in the private PAC learning setting, the classical and quantum sample complexities are equal, up to constant factors.

Non-interactive classical verification of quantum computation

Mon, 16 Nov 2020 00:00:00 +0000

In a recent breakthrough, Mahadev constructed an interactive protocol that enables a purely classical party to delegate any quantum computation to an untrusted quantum prover. In this work, we show that this same task can in fact be performed non-interactively and in zero-knowledge. Our protocols result from a sequence of significant improvements to the original four-message protocol of Mahadev. We begin by making the first message instance-independent and moving it to an offline setup phase. We then establish a parallel repetition theorem for the resulting three-message protocol, with an asymptotically optimal rate. This, in turn, enables an application of the Fiat-Shamir heuristic, eliminating the second message and giving a non-interactive protocol. Finally, we employ classical non-interactive zero-knowledge (NIZK) arguments and classical fully homomorphic encryption (FHE) to give a zero-knowledge variant of this construction. This yields the first purely classical NIZK argument system for QMA, a quantum analogue of NP. We establish the security of our protocols under standard assumptions in quantum-secure cryptography. Specifically, our protocols are secure in the Quantum Random Oracle Model, under the assumption that Learning with Errors is quantumly hard. The NIZK construction also requires circuit-private FHE.