arXiv subjects

Florian Speelman

Publications and source records attributed to Florian Speelman.

At least 37 records · Page 2Linked to original sources

arXiv2023

Security of a Continuous-Variable based Quantum Position Verification Protocol

In this work we study quantum position verification with continuous-variable quantum states. In contrast to existing discrete protocols, we present and analyze a protocol that utilizes coherent states and its properties. Compared to discrete-variable photonic states, coherent states offer practical advantages since they can be efficiently prepared and manipulated with current technology. We prove security of the protocol against any unentangled attackers via entropic uncertainty relations, showing that the adversary has more uncertainty than the honest prover about the correct response as long as the noise in the quantum channel is below a certain threshold. Additionally, we show that attackers who pre-share one continuous-variable EPR pair can break the protocol.

quant-ph↗

arXiv2023

A single-qubit position verification protocol that is secure against multi-qubit attacks

The position of a device or agent is an important security credential in today's society, both online and in the real world. Unless in direct proximity, however, the secure verification of a position is impossible without further assumptions. This is true classically, but also in any future quantum-equipped communications infrastructure. We show in this work that minimal quantum resources, in the form of a single qubit, combined with classical communication are sufficient to thwart quantum adversaries that pretend to be at a specific position and have the ability to coordinate their action with entanglement. More precisely, we show that the adversaries using an increasing amount of entanglement can be combatted solely by increasing the number of classical bits used in the protocol. The presented protocols are noise-robust and within reach of current quantum technology.

quant-ph↗

arXiv2022

An efficient combination of quantum error correction and authentication

When sending quantum information over a channel, we want to ensure that the message remains intact. Quantum error correction and quantum authentication both aim to protect (quantum) information, but approach this task from two very different directions: error-correcting codes protect against probabilistic channel noise and are meant to be very robust against small errors, while authentication codes prevent adversarial attacks and are designed to be very sensitive against any error, including small ones. In practice, when sending an authenticated state over a noisy channel, one would have to wrap it in an error-correcting code to counterbalance the sensitivity of the underlying authentication scheme. We study the question of whether this can be done more efficiently by combining the two functionalities in a single code. To illustrate the potential of such a combination, we design the threshold code, a modification of the trap authentication code which preserves that code's authentication properties, but which is naturally robust against depolarizing channel noise. We show that the threshold code needs polylogarithmically fewer qubits to achieve the same level of security and robustness, compared to the naive composition of the trap code with any concatenated CSS code. We believe our analysis opens the door to combining more general error-correction and authentication codes, which could improve the practicality of the resulting scheme.

quant-ph↗

arXiv2022

Towards practical and error-robust quantum position verification

Loss of inputs can be detrimental to the security of quantum position verification (QPV) protocols, as it may allow attackers to not answer on all played rounds, but only on those they perform well on. In this work, we study \textit{loss-tolerant} QPV protocols. We propose a new fully loss-tolerant protocol QPV$_{\textsf{SWAP}}$, based on the SWAP test, with several desirable properties. The task of the protocol, which could be implemented using only a single beam splitter and two detectors, is to estimate the overlap between two input states. By formulating possible attacks as a semi-definite program (SDP), we prove full loss tolerance against unentangled attackers restricted to local operations and classical communication, and show that the attack probability decays exponentially under parallel repetition of rounds. We show that the protocol remains secure even if unentangled attackers are allowed to quantum communicate, making our protocol the first fully loss-tolerant protocol with this property. A detailed analysis under experimental conditions is conducted, showing that QPV$_{\textsf{SWAP}}$ remains fairly robust against equipment errors. We identify a necessary condition for security with errors and simulate one instance of our protocol with currently realistic experimental parameters, gathering that an attack success probability of $\leq10^{-6}$ can be achieved by collecting just a few hundred conclusive protocol rounds.

quant-ph↗

arXiv2022

On the Role of Quantum Communication and Loss in Attacks on Quantum Position Verification

We study the role of quantum communication in attacks on quantum position verification. In this work, we construct the first known example of a QPV protocol that is provably secure against unentangled attackers restricted to classical communication, but can be perfectly attacked by local operations and a single round of simultaneous quantum communication indicating that allowing for quantum communication may break security. We also show that any protocol secure against classical communication can be transformed into a protocol secure against quantum communication. We further show, using arguments based on the monogamy of entanglement, that the task of Bell state discrimination cannot be done locally with a single round of quantum communication, not even probabilistically (when we allow attackers to say loss sometimes), making this the first fully loss-tolerant QPV task secure against quantum communication attacks. Finally, we observe that any multi-round QPV protocol can be attacked with a linear amount of entanglement if the loss is high enough.

quant-ph↗

arXiv2022

Matching Triangles and Triangle Collection: Hardness based on a Weak Quantum Conjecture

Classically, for many computational problems one can conclude time lower bounds conditioned on the hardness of one or more of key problems: k-SAT, 3SUM and APSP. More recently, similar results have been derived in the quantum setting conditioned on the hardness of k-SAT and 3SUM. This is done using fine-grained reductions, where the approach is to (1) select a key problem $X$ that, for some function $T$, is conjectured to not be solvable by any $O(T(n)^{1-ε})$ time algorithm for any constant $ε> 0$ (in a fixed model of computation), and (2) reduce $X$ in a fine-grained way to these computational problems, thus giving (mostly) tight conditional time lower bounds for them. Interestingly, for Delta-Matching Triangles and Triangle Collection, classical hardness results have been derived conditioned on hardness of all three mentioned key problems. More precisely, it is proven that an $n^{3-ε}$ time classical algorithm for either of these two graph problems would imply faster classical algorithms for k-SAT, 3SUM and APSP, which makes Delta-Matching Triangles and Triangle Collection worthwhile to study. In this paper, we show that an $n^{1.5-ε}$ time quantum algorithm for either of these two graph problems would imply faster quantum algorithms for k-SAT, 3SUM, and APSP. We first formulate a quantum hardness conjecture for APSP and then present quantum reductions from k-SAT, 3SUM, and APSP to Delta-Matching Triangles and Triangle Collection. Additionally, based on the quantum APSP conjecture, we are also able to prove quantum lower bounds for a matrix problem and many graph problems. The matching upper bounds follow trivially for most of them, except for Delta-Matching Triangles and Triangle Collection for which we present quantum algorithms that require careful use of data structures and Ambainis' variable time search.

cs.CC↗

arXiv2022

Solving correlation clustering with QAOA and a Rydberg qudit system: a full-stack approach

We study the correlation clustering problem using the quantum approximate optimization algorithm (QAOA) and qudits, which constitute a natural platform for such non-binary problems. Specifically, we consider a neutral atom quantum computer and propose a full stack approach for correlation clustering, including Hamiltonian formulation of the algorithm, analysis of its performance, identification of a suitable level structure for ${}^{87}{\rm Sr}$ and specific gate design. We show the qudit implementation is superior to the qubit encoding as quantified by the gate count. For single layer QAOA, we also prove (conjecture) a lower bound of $0.6367$ ($0.6699$) for the approximation ratio on 3-regular graphs. Our numerical studies evaluate the algorithm's performance by considering complete and Erdős-Rényi graphs of up to 7 vertices and clusters. We find that in all cases the QAOA surpasses the Swamy bound $0.7666$ for the approximation ratio for QAOA depths $p \geq 2$. Finally, by analysing the effect of errors when solving complete graphs we find that their inclusion severely limits the algorithm's performance.

quant-ph↗

arXiv2022

Memory Compression with Quantum Random-Access Gates

In the classical RAM, we have the following useful property. If we have an algorithm that uses $M$ memory cells throughout its execution, and in addition is sparse, in the sense that, at any point in time, only $m$ out of $M$ cells will be non-zero, then we may "compress" it into another algorithm which uses only $m \log M$ memory and runs in almost the same time. We may do so by simulating the memory using either a hash table, or a self-balancing tree. We show an analogous result for quantum algorithms equipped with quantum random-access gates. If we have a quantum algorithm that runs in time $T$ and uses $M$ qubits, such that the state of the memory, at any time step, is supported on computational-basis vectors of Hamming weight at most $m$, then it can be simulated by another algorithm which uses only $O(m \log M)$ memory, and runs in time $\tilde O(T)$. We show how this theorem can be used, in a black-box way, to simplify the presentation in several papers. Broadly speaking, when there exists a need for a space-efficient history-independent quantum data-structure, it is often possible to construct a space-inefficient, yet sparse, quantum data structure, and then appeal to our main theorem. This results in simpler and shorter arguments.

quant-ph↗

arXiv2021

Limits of quantum speed-ups for computational geometry and other problems: Fine-grained complexity via quantum walks

Many computational problems are subject to a quantum speed-up: one might find that a problem having an O(n^3)-time or O(n^2)-time classic algorithm can be solved by a known O(n^1.5)-time or O(n)-time quantum algorithm. The question naturally arises: how much quantum speed-up is possible? The area of fine-grained complexity allows us to prove optimal lower-bounds on the complexity of various computational problems, based on the conjectured hardness of certain natural, well-studied problems. This theory has recently been extended to the quantum setting, in two independent papers by Buhrman, Patro, and Speelman (arXiv:1911.05686), and by Aaronson, Chia, Lin, Wang, and Zhang (arXiv:1911.01973). In this paper, we further extend the theory of fine-grained complexity to the quantum setting. A fundamental conjecture in the classical setting states that the 3SUM problem cannot be solved by (classical) algorithms in time O(n^{2-a}), for any a>0. We formulate an analogous conjecture, the Quantum-3SUM-Conjecture, which states that there exist no sublinear O(n^{1-b})-time quantum algorithms for the 3SUM problem. Based on the Quantum-3SUM-Conjecture, we show new lower-bounds on the time complexity of quantum algorithms for several computational problems. Most of our lower-bounds are optimal, in that they match known upper-bounds, and hence they imply tight limits on the quantum speedup that is possible for these problems.

quant-ph↗

arXiv2021

A Guide for New Program Committee Members at Theoretical Computer Science Conferences

In theoretical computer science, conferences play an important role in the scientific process. The decisions whether to accept or reject articles is taken by the program committee (PC) members. Serving on a PC for the first time can be a daunting experience. This guide will help new program-committee members to understand how the system works, and provide useful tips and guidelines. It discusses every phase of the paper-selection process, and the tasks associated to it.

cs.GL↗

arXiv2019

Asymptotic performance of port-based teleportation

Quantum teleportation is one of the fundamental building blocks of quantum Shannon theory. While ordinary teleportation is simple and efficient, port-based teleportation (PBT) enables applications such as universal programmable quantum processors, instantaneous non-local quantum computation and attacks on position-based quantum cryptography. In this work, we determine the fundamental limit on the performance of PBT: for arbitrary fixed input dimension and a large number $N$ of ports, the error of the optimal protocol is proportional to the inverse square of $N$. We prove this by deriving an achievability bound, obtained by relating the corresponding optimization problem to the lowest Dirichlet eigenvalue of the Laplacian on the ordered simplex. We also give an improved converse bound of matching order in the number of ports. In addition, we determine the leading-order asymptotics of PBT variants defined in terms of maximally entangled resource states. The proofs of these results rely on connecting recently-derived representation-theoretic formulas to random matrix theory. Along the way, we refine a convergence result for the fluctuations of the Schur-Weyl distribution by Johansson, which might be of independent interest.

quant-ph↗

arXiv2019

The Quantum Strong Exponential-Time Hypothesis

The strong exponential-time hypothesis (SETH) is a commonly used conjecture in the field of complexity theory. It states that CNF formulas cannot be analyzed for satisfiability with a speedup over exhaustive search. This hypothesis and its variants gave rise to a fruitful field of research, fine-grained complexity, obtaining (mostly tight) lower bounds for many problems in P whose unconditional lower bounds are hard to find. In this work, we introduce a framework of Quantum Strong Exponential-Time Hypotheses, as quantum analogues to SETH. Using the QSETH framework, we are able to translate quantum query lower bounds on black-box problems to conditional quantum time lower bounds for many problems in BQP. As an example, we illustrate the use of the QSETH by providing a conditional quantum time lower bound of $Ω(n^{1.5})$ for the Edit Distance problem. We also show that the $n^2$ SETH-based lower bound for a recent scheme for Proofs of Useful Work, based on the Orthogonal Vectors problem holds for quantum computation assuming QSETH, maintaining a quadratic gap between verifier and prover.

quant-ph↗

arXiv2018

Round elimination in exact communication complexity

We study two basic graph parameters, the chromatic number and the orthogonal rank, in the context of classical and quantum exact communication complexity. In particular, we consider two types of communication problems that we call promise equality and list problems. For both of these, it was already known that the one-round classical and one-round quantum complexities are characterized by the chromatic number and orthogonal rank of a certain graph, respectively. In a promise equality problem, Alice and Bob must decide if their inputs are equal or not. We prove that classical protocols for such problems can always be reduced to one-round protocols with no extra communication. In contrast, we give an explicit instance of a promise equality problem that exhibits an exponential gap between the one- and two-round exact quantum communication complexities. Whereas the chromatic number thus fully captures the complexity of promise equality problems, the hierarchy of "quantum chromatic numbers" (starting with the orthogonal rank) giving the quantum communication complexity for every fixed number of communication rounds turns out to enjoy a much richer structure. In a list problem, Bob gets a subset of some finite universe, Alice gets an element from Bob's subset, and their goal is for Bob to learn which element Alice was given. The best general lower bound (due to Orlitsky) and upper bound (due to Naor, Orlitsky, and Shor) on the classical communication complexity of such problems differ only by a constant factor. We exhibit an example showing that, somewhat surprisingly, the four-round protocol used in the bound of Naor et al. can in fact be optimal. Finally, we pose a conjecture on the orthogonality rank of a certain graph whose truth would imply an intriguing impossibility of round elimination in quantum protocols for list problems, something that works trivially in the classical case.

quant-ph↗

arXiv2018

Quantum ciphertext authentication and key recycling with the trap code

We investigate quantum authentication schemes constructed from quantum error-correcting codes. We show that if the code has a property called purity testing, then the resulting authentication scheme guarantees the integrity of ciphertexts, not just plaintexts. On top of that, if the code is strong purity testing, the authentication scheme also allows the encryption key to be recycled, partially even if the authentication rejects. Such a strong notion of authentication is useful in a setting where multiple ciphertexts can be present simultaneously, such as in interactive or delegated quantum computation. With these settings in mind, we give an explicit code (based on the trap code) that is strong purity testing but, contrary to other known strong-purity-testing codes, allows for natural computation on ciphertexts.

quant-ph↗

arXiv2017

Quantum Fully Homomorphic Encryption With Verification

Fully-homomorphic encryption (FHE) enables computation on encrypted data while maintaining secrecy. Recent research has shown that such schemes exist even for quantum computation. Given the numerous applications of classical FHE (zero-knowledge proofs, secure two-party computation, obfuscation, etc.) it is reasonable to hope that quantum FHE (or QFHE) will lead to many new results in the quantum setting. However, a crucial ingredient in almost all applications of FHE is circuit verification. Classically, verification is performed by checking a transcript of the homomorphic computation. Quantumly, this strategy is impossible due to no-cloning. This leads to an important open question: can quantum computations be delegated and verified in a non-interactive manner? In this work, we answer this question in the affirmative, by constructing a scheme for QFHE with verification (vQFHE). Our scheme provides authenticated encryption, and enables arbitrary polynomial-time quantum computations without the need of interaction between client and server. Verification is almost entirely classical; for computations that start and end with classical states, it is completely classical. As a first application, we show how to construct quantum one-time programs from classical one-time programs and vQFHE.

quant-ph↗

arXiv2016

Instantaneous non-local computation of low T-depth quantum circuits

Instantaneous non-local quantum computation requires multiple parties to jointly perform a quantum operation, using pre-shared entanglement and a single round of simultaneous communication. We study this task for its close connection to position-based quantum cryptography, but it also has natural applications in the context of foundations of quantum physics and in distributed computing. The best known general construction for instantaneous non-local quantum computation requires a pre-shared state which is exponentially large in the number of qubits involved in the operation, while efficient constructions are known for very specific cases only. We partially close this gap by presenting new schemes for efficient instantaneous non-local computation of several classes of quantum circuits, using the Clifford+T gate set. Our main result is a protocol which uses entanglement exponential in the T-depth of a quantum circuit, able to perform non-local computation of quantum circuits with a (poly-)logarithmic number of layers of T gates with quasi-polynomial entanglement. Our proofs combine ideas from blind and delegated quantum computation with the garden-hose model, a combinatorial model of communication complexity which was recently introduced as a tool for studying certain schemes for quantum position verification. As an application of our results, we also present an efficient attack on a recently-proposed scheme for position verification by Chakraborty and Leverrier.

quant-ph↗

arXiv2016

Quantum homomorphic encryption for polynomial-sized circuits

We present a new scheme for quantum homomorphic encryption which is compact and allows for efficient evaluation of arbitrary polynomial-sized quantum circuits. Building on the framework of Broadbent and Jeffery and recent results in the area of instantaneous non-local quantum computation, we show how to construct quantum gadgets that allow perfect correction of the errors which occur during the homomorphic evaluation of T gates on encrypted quantum data. Our scheme can be based on any classical (leveled) fully homomorphic encryption (FHE) scheme and requires no computational assumptions besides those already used by the classical scheme. The size of our quantum gadget depends on the space complexity of the classical decryption function -- which aligns well with the current efforts to minimize the complexity of the decryption function. Our scheme (or slight variants of it) offers a number of additional advantages such as ideal compactness, the ability to supply gadgets "on demand", circuit privacy for the evaluator against passive adversaries, and a three-round scheme for blind delegated quantum computation which puts only very limited demands on the quantum abilities of the client.

quant-ph↗

arXiv2015

Quantum communication complexity advantage implies violation of a Bell inequality

We obtain a general connection between a quantum advantage in communication complexity and non-locality. We show that given any protocol offering a (sufficiently large) quantum advantage in communication complexity, there exists a way of obtaining measurement statistics which violate some Bell inequality. Our main tool is port-based teleportation. If the gap between quantum and classical communication complexity can grow arbitrarily large, the ratio of the quantum value to the classical value of the Bell quantity becomes unbounded with the increase in the number of inputs and outputs.

quant-ph↗