Classical Oracle Research Articles

A distribution testing oracle separation between QMA and QCMA

It is a long-standing open question in quantum complexity theory whether the definition of non−deterministic quantum computation requires quantum witnesses (QMA) or if classical witnesses suffice (QCMA). We make progress on this question by constructing a randomized classical oracle separating the respective computational complexity classes. Previous separations \cite{ak-oracle}, \cite{fefferman-kimmel} required a quantum unitary oracle. The separating problem is deciding whether a distribution supported on regular un-directed graphs either consists of multiple connected components (yes instances) or consists of one expanding connected component (no instances) where the graph is given in an adjacency-list format by the oracle. Therefore, the oracle is a distribution over n-bit boolean functions.

Cryptanalysis of the «Vershyna» digital signature algorithm

The CRYSTALS-Dilithium digital signature algorithm, which was selected as the prototype of the new «Vershyna» digital signature algorithm, is analyzed in this paper. The characteristics of the National Digital Signature Standard Project and the construction of the «Vershyna» algorithm are also presented. During the analysis of the project, the predicted number of iterations that the algorithm must perform to create the correct signature was calculated. In addition, basic theoretical information about the structure of Fiat-Shamir with aborts and its security in quantum and classical models oracle models is also provided. We obtain our own results on the resistance of the «Vershyna» algorithm to the attack without the use of a message in classical and quantum oracle models. The resistance of the «Vershyna» algorithm to a key recovery attack is based on the assumption of the hardness of the MLWE~problem, and the resistance to existential signature forgery is based on the assumption of the hardness of the MSIS~problem. In this work, the expected level of hardness of SIS~and LWE~problems is calculated, to which there are reductions from MSIS~and MLWE~problems.

Tightly Secure PKE Combiner in the Quantum Random Oracle Model

The development of increasingly sophisticated quantum computers poses a long-term threat to current cryptographic infrastructure. This has spurred research into both quantum-resistant algorithms and how to safely transition real-world implementations and protocols to quantum-resistant replacements. This transition is likely to be a gradual process due to both the complexity and cost associated with transitioning. One method to ease the transition is the use of classical–quantum hybrid schemes, which provide security against both classical and quantum adversaries. We present a new combiner for creating hybrid encryption schemes directly from traditional encryption schemes. Our construction is the only existing proposal in the literature with IND-CCA-security in the classical and quantum random oracle models, respectively.

Quantum computing without quantum computers: Database search and data processing using classical wave superposition

Quantum computers are proven to be more efficient at solving a specific class of problems compared to traditional digital computers. Superposition of states and quantum entanglement are the two key ingredients that make quantum computing so powerful. However, not all quantum algorithms require quantum entanglement (e.g., search through an unsorted database). Is it possible to utilize classical wave superposition to speed up database searching as much as by using quantum computers? There were several attempts to mimic quantum computers using classical waves. It was concluded that the use of classical wave superposition comes with the cost of an exponential increase in resources. In this work, we consider the feasibility of building classical wave-based devices able to provide fundamental speedup over digital counterparts without the exponential overhead. We present experimental data on database searching through a magnetic database using spin wave superposition. The results demonstrate the same speedup as expected for quantum computers. Also, we present examples of numerical modeling demonstrating classical wave interference for period finding. This approach may not compete with quantum computers with efficiency but outperform classical digital computers. We argue that classical wave-based devices can perform some of the quantum algorithms with the same efficiency as quantum computers as long as quantum entanglement is not required.

Practically secure quantum position verification

We discuss quantum position verification (QPV) protocols in which the verifiers create and send single-qubit states to the prover. QPV protocols using single-qubit states are known to be insecure against adversaries that share a small number of entangled qubits. We introduce QPV protocols that are practically secure: they only require single-qubit states from each of the verifiers, yet their security is broken if the adversaries sharing an impractically large number of entangled qubits employ teleportation-based attacks. These protocols are a modification of known QPV protocols in which we include a classical random oracle without altering the amount of quantum resources needed by the verifiers. We present a cheating strategy that requires a number of entangled qubits shared among the adversaries that grows exponentially with the size of the classical input of the random oracle.

Early stopping for statistical inverse problems via truncated SVD estimation

We consider truncated SVD (or spectral cut-off, projection) estimators for a prototypical statistical inverse problem in dimension $D$. Since calculating the singular value decomposition (SVD) only for the largest singular values is much less costly than the full SVD, our aim is to select a data-driven truncation level $\widehat{m}\in \{1,\ldots ,D\}$ only based on the knowledge of the first $\widehat{m}$ singular values and vectors. We analyse in detail whether sequential early stopping rules of this type can preserve statistical optimality. Information-constrained lower bounds and matching upper bounds for a residual based stopping rule are provided, which give a clear picture in which situation optimal sequential adaptation is feasible. Finally, a hybrid two-step approach is proposed which allows for classical oracle inequalities while considerably reducing numerical complexity.

Sharp Threshold Detection Based on Sup-Norm Error Rates in High-Dimensional Models

We propose a new estimator, the thresholded scaled Lasso, in high-dimensional threshold regressions. First, we establish an upper bound on the ℓ∞ estimation error of the scaled Lasso estimator of Lee, Seo, and Shin. This is a nontrivial task as the literature on high-dimensional models has focused almost exclusively on ℓ1 and ℓ2 estimation errors. We show that this sup-norm bound can be used to distinguish between zero and nonzero coefficients at a much finer scale than would have been possible using classical oracle inequalities. Thus, our sup-norm bound is tailored to consistent variable selection via thresholding. Our simulations show that thresholding the scaled Lasso yields substantial improvements in terms of variable selection. Finally, we use our estimator to shed further empirical light on the long-running debate on the relationship between the level of debt (public and private) and GDP growth. Supplementary materials for this article are available online.

Computation in generalised probabilisitic theories

From the general difficulty of simulating quantum systems using classical systems, and in particular the existence of an efficient quantum algorithm for factoring, it is likely that quantum computation is intrinsically more powerful than classical computation. At present, the best upper bound known for the power of quantum computation is that , where is a classical complexity class (known to be included in , hence ). This work investigates limits on computational power that are imposed by simple physical, or information theoretic, principles. To this end, we define a circuit-based model of computation in a class of operationally-defined theories more general than quantum theory, and ask: what is the minimal set of physical assumptions under which the above inclusions still hold? We show that given only an assumption of tomographic locality (roughly, that multipartite states and transformations can be characterized by local measurements), efficient computations are contained in . This inclusion still holds even without assuming a basic notion of causality (where the notion is, roughly, that probabilities for outcomes cannot depend on future measurement choices). Following Aaronson, we extend the computational model by allowing post-selection on measurement outcomes. Aaronson showed that the corresponding quantum complexity class, , is equal to . Given only the assumption of tomographic locality, the inclusion in still holds for post-selected computation in general theories. Hence in a world with post-selection, quantum theory is optimal for computation in the space of all operational theories. We then consider whether one can obtain relativized complexity results for general theories. It is not obvious how to define a sensible notion of a computational oracle in the general framework that reduces to the standard notion in the quantum case. Nevertheless, it is possible to define computation relative to a ‘classical oracle’. Then, we show there exists a classical oracle relative to which efficient computation in any theory satisfying the causality assumption does not include .

Faster Together: Collective Quantum Search

Joining independent quantum searches provides novel collective modes of quantum search (merging) by utilizing the algorithm’s underlying algebraic structure. If n quantum searches, each targeting a single item, join the domains of their classical oracle functions and sum their Hilbert spaces (merging), instead of acting independently (concatenation), then they achieve a reduction of the search complexity by factor O(√n).

Sharp Threshold Detection based on Sup-Norm Error Rates in High-Dimensional Models

We propose a new estimator, the thresholded scaled Lasso, in high dimensional threshold regressions. First, we establish an upper bound on the l∞ estimation error of the scaled Lasso estimator of Lee et al. (2012). This is a non-trivial task as the literature on high-dimensional models has focused almost exclusively on l1 and l2 estimation errors. We show that this sup-norm bound can be used to distinguish between zero and non-zero coefficients at a much finer scale than would have been possible using classical oracle inequalities. Thus, our sup-norm bound is tailored to consistent variable selection via thresholding. Our simulations show that thresholding the scaled Lasso yields substantial improvements in terms of variable selection. Finally, we use our estimator to shed further empirical light on the long running debate on the relationship between the level of debt (public and private) and GDP growth.

Quantum Money from Hidden Subspaces

Forty years ago, Wiesner pointed out that quantum mechanics raises the striking possibility of money that cannot be counterfeited according to the laws of physics. We propose the first quantum money scheme that is (1) public-key, meaning that anyone can verify a banknote as genuine, not only the bank that printed it, and (2) cryptographically secure, under a hardness assumption that has nothing to do with quantum money. Our scheme is based on hidden subspaces, encoded as the zero-sets of random multivariate polynomials. A main technical advance is to show that the black-box version of our scheme, where the polynomials are replaced by classical oracles, is unconditionally secure. Previously, such a result had only been known relative to a quantum oracle (and even there, the proof was never published). Even in Wiesner's original setting -- quantum money that can only be verified by the bank -- we are able to use our techniques to patch a major security hole in Wiesner's scheme. We give the first private-key quantum money scheme that allows unlimited verifications and that remains unconditionally secure, even if the counterfeiter can interact adaptively with the bank. Our money scheme is simpler than previous public-key quantum money schemes, including a knot-based scheme of Farhi et al. The verifier needs to perform only two tests, one in the standard basis and one in the Hadamard basis -- matching the original intuition for quantum money, based on the existence of complementary observables. Our security proofs use a new variant of Ambainis's quantum adversary method, and several other tools that might be of independent interest.

A Quantum Algorithm Detecting Concentrated Maps.

We consider an arbitrary mapping f: {0, …, N − 1} → {0, …, N − 1} for N = 2n, n some number of quantum bits. Using N calls to a classical oracle evaluating f(x) and an N-bit memory, it is possible to determine whether f(x) is one-to-one. For some radian angle 0 ≤ θ ≤ π/2, we say f(x) is θ − concentrated if and only if for some given ψ0 and any 0 ≤ x ≤ N − 1. We present a quantum algorithm that distinguishes a θ-concentrated f(x) from a one-to-one f(x) in O(1) calls to a quantum oracle function Uf with high probability. For 0 < θ < 0.3301 rad, the quantum algorithm outperforms random (classical) evaluation of the function testing for dispersed values (on average). Maximal outperformance occurs at rad.

Quantum Versus Classical Proofs and Advice

This paper studies whether proofs are more powerful than classical proofs, or in complexity terms, whether QMA=QCMA. We prove three results about this question. First, we give a quantum oracle between QMA and QCMA. More concretely, we show that any algorithm needs $\Omega(\sqrt{2^n/(m+1)})$ queries to find an $n$-qubit marked state $\lvert\psi\rangle$, even if given an $m$-bit classical description of $\lvert\psi\rangle$ together with a black box that recognizes $\lvert\psi\rangle$. Second, we give an explicit QCMA protocol that nearly achieves this lower bound. Third, we show that, in the one previously-known case where proofs seemed to provide an exponential advantage, classical proofs are basically just as powerful. In particular, Watrous gave a QMA protocol for verifying non-membership in finite groups. Under plausible group-theoretic assumptions, we give a QCMA protocol for the same problem. Even with no assumptions, our protocol makes only polynomially many queries to the group oracle. We end with some conjectures about versus classical oracles, and about the possibility of a classical oracle separation between QMA and QCMA.

QUANTUM-CLASSICAL CORRESPONDENCE IN THE ORACLE MODEL OF COMPUTATION

The oracle model of computation is believed to allow a rigorous proof of quantum over classical computational superiority. Since quantum and classical oracles are essentially different, a correspondence principle is commonly implicitly used as a platform for comparison of oracle complexity. Here, we question the grounds on which this correspondence is based. Obviously, results on quantum speed-up depend on the chosen correspondence. So, we introduce the notion of genuine quantum speed-up which can serve as a tool for reliable comparison of quantum versus classical complexity, independent of the chosen correspondence principle.

An upper bound on the rate of information transfer by Grover's oracle

Grover discovered a quantum algorithm for identifying a target element in an unstructured search universe of N items in approximately \(\pi/4 \sqrt{N}\) queries to a quantum oracle. For classical search using a classical oracle, the search complexity is of order N/2 queries since on average half of the items must be searched. In work preceding Grover’s, Bennett et al. had shown that no quantum algorithm can solve the search problem in fewer than \(O(\sqrt{N})\) queries. Thus, Grover’s algorithm has optimal order of complexity. Here, we present an information-theoretic analysis of Grover’s algorithm and show that the square-root speed-up by Grover’s algorithm is the best possible by any algorithm using the same quantum oracle.

Fielding's Fallen Oracles: Print Culture and the Elusiveness of Common Sense

Laura McGrane is assistant professor of English and American literature at Haverford College. She is editor, with Bruce Thompson and Ryan Johnson, of Critical History: The Career of Ian Watt (2000) in the Stanford Humanities Review series and is at work on a book titled Hollowed Voices: Reformulations of the Classical Oracle in Eighteenth-Century British Print Culture.

General Bounds for Quantum Biased Oracles

An oracle with bias e is an oracle that answers queries correctly with a probability of at least 1/2+e. In this paper, we study the upper and lower bounds of quantum query complexity of oracles with bias e. For general upper bounds, we show that for any quantum algorithm solving some problem with high probability using T queries of perfect quantum oracles, i.e., oracles with e =1/2, there exists a quantum algorithm solving the same problem, also with high probability, using O(T/e) queries of the corresponding biased quantum oracles. As corollaries we can show robust quantum algorithms and gaps between biased quantum and classical oracles, e.g., by showing a problem where the quantum query complexity is O(N/e) but the classical query complexity is lower bounded by Ω(N logN/e2). For general lower bounds, we generalize Ambainis' quantum adversary argument to biased quantum oracles and obtain the first lower bounds with explicit bias factor. As one of its applications we can provide another proof of the optimality of quantum algorithms for the so-called quantum Goldreich-Levin problem which was proved before by Adcock, et al. using different and more complicated methods.

The rhetoric of Romantic prophecy

The Romantic era in England and Germany saw a sudden renewal of prophetic modes of writing. Biblical prophecy and, to a lesser extent, classical oracle again became viable models for poetry and even for journalistic prose. Notably, this development arose out of the new-found freedom of biblical interpretation that began in the mid-eighteenth century, as the Bible was increasingly seen to be a literary and mythical text. Taking Walter Benjamin's thinking about history as a point of departure, the author shows how the model for Romantic prophecy emerges less as a prediction of the future than as a call to change in the present, even as it quotes, at key turns, texts from the past. After surveying developments in eighteenth-century biblical hermeneutics, as well as the numerous instances of prophetic eruption in Romantic poetry, the book culminates in close readings of works by Blake, Holderlin, and Coleridge. Each of these writers interpreted the Bible in strong, variously radical and conservative ways, and each reworked prophetic texts in often startling fashion. The author's reading of Blake focuses on the complex temporal and rhetorical dynamics at work in a prophetic tradition, with attention paid to the key mediating figure of Milton. The chapter on Holderlin investigates the truth-claim of poetry and the consequences of Holderlin's insight into the necessarily figural character of poetry. The analysis of Coleridge correlates his theory of allegory and symbol with his theory and practice of political writing, which often relies on mobilizing prophetic authority. Together, the readings force us to reexamine the claims and practices of Romantic poets and thinkers and their ideas and ideologies, not without engendering some allegorical resonance with issues in our own time.

Comparison of quantum oracles

A standard quantum oracle $S_f$ for a general function $f: Z_N \to Z_N $ is defined to act on two input states and return two outputs, with inputs $\ket{i}$ and $\ket{j}$ ($i,j \in Z_N $) returning outputs $\ket{i}$ and $\ket{j \oplus f(i)}$. However, if $f$ is known to be a one-to-one function, a simpler oracle, $M_f$, which returns $\ket{f(i)}$ given $\ket{i}$, can also be defined. We consider the relative strengths of these oracles. We define a simple promise problem which minimal quantum oracles can solve exponentially faster than classical oracles, via an algorithm which cannot be naively adapted to standard quantum oracles. We show that $S_f$ can be constructed by invoking $M_f$ and $(M_f)^{-1}$ once each, while $\Theta(\sqrt{N})$ invocations of $S_f$ and/or $(S_f)^{-1}$ are required to construct $M_f$.