Harry Buhrman

Signal loss poses a significant threat to the security of quantum cryptography when the chosen protocol lacks loss-tolerance. In quantum position verification (QPV) protocols, even relatively small loss rates can compromise security. The goal is thus to find protocols that remain secure under practically achievable loss rates. In this work, we modify the usual structure of QPV protocols and prove that this modification makes the potentially high transmission loss between the verifiers and the prover security-irrelevant for a class of protocols that includes a practically-interesting candidate protocol inspired by the BB84 protocol. This modification, which involves photon presence detection, a small time delay at the prover, and a commitment to play before proceeding, reduces the overall loss rate to just the prover’s laboratory. The adapted protocol then becomes a practically feasible QPV protocol with strong security guarantees, even against attackers using adaptive strategies. As the loss rate between the verifiers and prover is mainly dictated by the distance between them, secure QPV over longer distances becomes possible. We also show possible implementations of the required photon presence detection, making the adapted protocol a protocol that solves all major practical issues in QPV. Finally, we discuss experimental aspects and give parameter estimations.

The quantum analogue of the equality function, known as the quantum state identity problem, is the task of deciding whether n unknown quantum states are equal or unequal, given the promise that all states are either pairwise orthogonal or identical. Under the one-sided error requirement, it is known that the permutation test is optimal for this task, and for two input states this coincides with the well-known Swap test. Until now, the optimal measurement in the general two-sided error regime was unknown. Under more specific promises, the problem can be solved approximately or even optimally with simpler tests, such as the circle test. This work attempts to capture the underlying structure of (fine-grained formulations of) the quantum state identity problem. Using tools from semi-definite programming and representation theory, we (i) give an optimal test for any input distribution without the one-sided error requirement by writing the problem as an SDP, giving the exact solutions to the primal and dual programs and showing that the two values coincide; (ii) propose a general G-test which uses an arbitrary subgroup G of S_n, giving an analytic expression of the performance of the specific test, and (iii) give an approximation of the permutation test using only a classical permutation and n−1 Swap tests.

We define a general formulation of quantum PCPs, which captures adaptivity and multiple unentangled provers, and give a detailed construction of the quantum reduction to a local Hamiltonian with a constant promise gap. The reduction turns out to be a versatile subroutine to prove properties of quantum PCPs, allowing us to show: (i) Non-adaptive quantum PCPs can simulate adaptive quantum PCPs when the number of proof queries is constant. In fact, this can even be shown to hold when the non-adaptive quantum PCP picks the proof indices simply uniformly at random from a subset of all possible index combinations, answering an open question by Aharonov, Arad, Landau and Vazirani (STOC '09). (ii) If the q-local Hamiltonian problem with constant promise gap can be solved in 𝖰𝖢𝖬𝖠, then 𝖰𝖯𝖢𝖯[q] is in 𝖰𝖢𝖬𝖠 for any constant q. (iii) If 𝖰𝖬𝖠(k) has a quantum PCP for any k=poly(n), then 𝖰𝖬𝖠(2) = 𝖰𝖬𝖠, connecting two of the longest-standing open problems in quantum complexity theory. Moreover, we also show that there exists (quantum) oracles relative to which certain quantum PCP statements are false. Hence, any attempt to prove the quantum PCP conjecture requires, just as was the case for the classical PCP theorem, (quantumly) non-relativizing techniques.

Abstract Majority vote is a basic method for amplifying correct outcomes that is widely used in computer science and beyond. It can, for example, be used to amplify the correctness of a quantum device whose output is classical. However, when the output of a device is a quantum state, it is not apriori clear how to implement an analogous \emph{quantum} majority vote. To this end, we consider an extension of majority vote to quantum inputs and outputs: given a product state of the form $\ket{\phi_1, \phi_2, \dotsc ,\phi_n}$ where each qubit $\ket{\phi_i}$ is in one of two orthogonal states $\ket{\psi_0}$ or $\ket{\psi_1}$, output the majority state $\ket{\psi_0}$ or $\ket{\psi_1}$. We provide an optimal algorithm for this problem that achieves worst-case fidelity of $1/2 + \Theta(1/\sqrt{n})$. Under the promise that at least $2/3$ of the qubits are in the majority state, the fidelity increases to $1 - \Theta(1/n)$ and approaches one in the limit. More generally, we initiate the study of covariant and symmetric Boolean functions $f: \set{0,1^n} \to \set{0,1}$ with quantum inputs and outputs. We provide a simple linear program of size roughly $n/2$ for computing the optimal worst-case fidelity and show that a generalization of our algorithm is optimal for computing $f$. Our algorithm has complexity $O(n^4 \log n)$ where $n$ is the number of qubits.

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 $\Omega(n^{1.5})$ for the Longest Common Subsequence and Edit Distance problems. 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.

Most algorithms in Quantum Computation are developed in the black box setting. This is the setting where we are interested in determining the property of some function f : {0,1}^n → {0,1}. The algorithms are geared to determine this property with as few applications---black-box calls---of f as possible.

In this work, we study loss-tolerant quantum position verification (QPV) protocols. We propose a new fully loss-tolerant protocol, based on the SWAP test, with several desirable properties. The task of the protocol, which can 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 (LOCC), and additionally show that the attack probability decays exponentially under parallel repetition of rounds. Furthermore, we investigate the role of loss and quantum communication attacks in QPV in general. A protocol that is provably secure against unentangled attackers restricted to LOCC, but can be perfectly attacked by local operations and a single round of simultaneous quantum communication, is constructed. However, we show that any protocol secure against classical communication can be transformed into a protocol secure against quantum communication. Finally, we observe that any QPV protocol can be attacked with a linear amount of entanglement if the loss is high enough.