39 research outputs found
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
Single-qubit loss-tolerant quantum position verification protocol secure against entangled attackers
Protocols for quantum position verification (QPV) which combine classical and
quantum information are insecure in the presence of loss. We study the exact
loss-tolerance of the most popular protocol for QPV, which is based on BB84
states, and generalizations of this protocol. By bounding the winning
probabilities of a variant of the monogamy-of-entanglement game using
semidefinite programming (SDP), we find tight bounds for the relation between
loss and error for these extended non-local games.
These new bounds enable the usage of QPV protocols using more-realistic
experimental parameters. We show how these results transfer to the variant
protocol which combines bits of classical information with a single qubit,
thereby exhibiting a protocol secure against a linear amount of entanglement
(in the classical information ) even in the presence of a moderate amount of
photon loss. Moreover, this protocol stays secure even if the photon encoding
the qubit travels arbitrarily slow in an optical fiber. We also extend this
analysis to the case of more than two bases, showing even stronger
loss-tolerance for that case.
Finally, since our semi-definite program bounds a monogamy-of-entanglement
game, we describe how they can also be applied to improve the analysis of
one-sided device-independent QKD protocols
Position-based cryptography: Single-qubit protocol secure against multi-qubit attacks
While it is known that unconditionally secure position-based cryptography is
impossible both in the classical and the quantum setting, it has been shown
that some quantum protocols for position verification are secure against
attackers which share a quantum state of bounded dimension. In this work, we
consider the security of two protocols for quantum position verification that
combine a single qubit with classical strings of total length : The qubit
routing protocol, where the classical information prescribes the qubit's
destination, and a variant of the BB84-protocol for position verification,
where the classical information prescribes in which basis the qubit should be
measured. We show that either protocol is secure for a randomly chosen function
if each of the attackers holds at most qubits. With this, we show for
the first time that there exists a quantum position verification protocol where
the ratio between the quantum resources an honest prover needs and the quantum
resources the attackers need to break the protocol is unbounded. The verifiers
need only increase the amount of classical resources to force the attackers to
use more quantum resources. Concrete efficient functions for both protocols are
also given -- at the expense of a weaker but still unbounded ratio of quantum
resources for successful attackers. Finally, we show that both protocols are
robust with respect to noise, making them appealing for applications.Comment: 26 pages, 4 figures. Content significantly expanded. In particular,
we have added the function BB84 protocol and prove its security in Section 4.
Finally, we give lower bounds for concrete functions in Section
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
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 of ports, the error of the optimal protocol is
proportional to the inverse square of . 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.Comment: 68 pages, 4 figures; comments welcome! v2: minor fixes, added plots
comparing asymptotic expansions to exact formulas, code available at
https://github.com/amsqi/port-base
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 problem that exhibits an exponential gap between the one- and two-round exact quantum communication complexities. Whereas the chromatic number thus captures the complete 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 thus 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\u27s 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
QSETH strikes again: finer quantum lower bounds for lattice problem, strong simulation, hitting set problem, and more
While seemingly undesirable, it is not a surprising fact that there are
certain problems for which quantum computers offer no computational advantage
over their respective classical counterparts. Moreover, there are problems for
which there is no `useful' computational advantage possible with the current
quantum hardware. This situation however can be beneficial if we don't want
quantum computers to solve certain problems fast - say problems relevant to
post-quantum cryptography. In such a situation, we would like to have evidence
that it is difficult to solve those problems on quantum computers; but what is
their exact complexity?
To do so one has to prove lower bounds, but proving unconditional time lower
bounds has never been easy. As a result, resorting to conditional lower bounds
has been quite popular in the classical community and is gaining momentum in
the quantum community. In this paper, by the use of the QSETH framework
[Buhrman-Patro-Speelman 2021], we are able to understand the quantum complexity
of a few natural variants of CNFSAT, such as parity-CNFSAT or counting-CNFSAT,
and also are able to comment on the non-trivial complexity of
approximate-#CNFSAT; both of these have interesting implications about the
complexity of (variations of) lattice problems, strong simulation and hitting
set problem, and more.
In the process, we explore the QSETH framework in greater detail than was
(required and) discussed in the original paper, thus also serving as a useful
guide on how to effectively use the QSETH framework.Comment: 34 pages, 2 tables, 2 figure