We show how to translate a subset of RISC-V machine code compiled from a subset of C to quadratic unconstrained binary optimization (QUBO) models that may be solved by a quantum annealing machine: given a bound $n$, there is input $I$ to a program $P$ such that $P$ runs into a given program state $E$ executing no more than $n$ machine instructions if and only if the QUBO model of $P$ for $n$ evaluates to 0 on $I$. Thus, with more qubits on the machine than variables in the QUBO model, quantum annealing the model reaches 0 (ground) energy in constant time with high probability on some input $I$ that is part of the ground state if and only if $P$ runs into $E$ on $I$ executing no more than $n$ instructions. Translation takes $\mathcal{O}(n^2)$ time effectively turning a quantum annealer into a polynomial-time symbolic execution engine and bounded model checker, eliminating their path and state explosion problems. Here, we take advantage of the fact that any machine instruction may only increase the size of the program state by a constant amount of bits. Translation time comes down from $\mathcal{O}(n^2)$ to $\mathcal{O}(n)$ if memory consumption of $P$ is bounded by a constant, establishing a linear (quadratic) upper bound on quantum space, in number of qubits on a quantum annealer, in terms of algorithmic time (space) in classical computing. The construction provides a non-relativizing argument for $NP\subseteq BQP$, without violating the optimality of Grover's algorithm, also on gate-model quantum machines, and motivates a temporal and spatial metric of quantum advantage. Our prototypical open-source toolchain translates machine code that runs on real RISC-V hardware to models that can be solved by real quantum annealing hardware, as shown in our experiments.
相關內容
ACM/IEEE第23屆模型驅動工程語言和系統國際會議，是模型驅動軟件和系統工程的首要會議系列，由ACM-SIGSOFT和IEEE-TCSE支持組織。自1998年以來，模型涵蓋了建模的各個方面，從語言和方法到工具和應用程序。模特的參加者來自不同的背景，包括研究人員、學者、工程師和工業專業人士。MODELS 2019是一個論壇，參與者可以圍繞建模和模型驅動的軟件和系統交流前沿研究成果和創新實踐經驗。今年的版本將為建模社區提供進一步推進建模基礎的機會，并在網絡物理系統、嵌入式系統、社會技術系統、云計算、大數據、機器學習、安全、開源等新興領域提出建模的創新應用以及可持續性。
官網鏈接： ·
泛函
·
離散化
·
相同
·
Next
·
Quantum computing is evolving so quickly that forces us to revisit, rewrite, and update the basis of the theory. Basic Quantum Algorithms revisits the first quantum algorithms. It started in 1995 with Deutsch trying to evaluate a function at two domain points simultaneously. Then, Deutsch and Jozsa created in 1992 a quantum algorithm that determines whether a Boolean function is constant or balanced. In the next year, Bernstein and Vazirani realized that the same algorithm can be used to find a specific Boolean function in the set of linear Boolean functions. In 1994, Simon presented a new quantum algorithm that determines whether a function is one-to-one or two-to-one exponentially faster than any classical algorithm for the same problem. In the same year, Shor created two new quantum algorithms for factoring integers and calculating discrete logarithms, threatening the cryptography methods widely used nowadays. In 1995, Kitaev described an alternative version for Shor's algorithms that proved useful in many other applications. In the following year, Grover created a quantum search algorithm quadratically faster than its classical counterpart. In this work, all those remarkable algorithms are described in detail with a focus on the circuit model.
In the training of over-parameterized model functions via gradient descent, sometimes the parameters do not change significantly and remain close to their initial values. This phenomenon is called lazy training, and motivates consideration of the linear approximation of the model function around the initial parameters. In the lazy regime, this linear approximation imitates the behavior of the parameterized function whose associated kernel, called the tangent kernel, specifies the training performance of the model. Lazy training is known to occur in the case of (classical) neural networks with large widths. In this paper, we show that the training of geometrically local parameterized quantum circuits enters the lazy regime for large numbers of qubits. More precisely, we prove bounds on the rate of changes of the parameters of such a geometrically local parameterized quantum circuit in the training process, and on the precision of the linear approximation of the associated quantum model function; both of these bounds tend to zero as the number of qubits grows. We support our analytic results with numerical simulations.
In this paper, we study quantum algorithms for computing the exact value of the treewidth of a graph. Our algorithms are based on the classical algorithm by Fomin and Villanger (Combinatorica 32, 2012) that uses $O(2.616^n)$ time and polynomial space. We show three quantum algorithms with the following complexity, using QRAM in both exponential space algorithms: $\bullet$ $O(1.618^n)$ time and polynomial space; $\bullet$ $O(1.554^n)$ time and $O(1.452^n)$ space; $\bullet$ $O(1.538^n)$ time and space. In contrast, the fastest known classical algorithm for treewidth uses $O(1.755^n)$ time and space. The first two speed-ups are obtained in a fairly straightforward way. The first version uses additionally only Grover's search and provides a quadratic speedup. The second speedup is more time-efficient and uses both Grover's search and the quantum exponential dynamic programming by Ambainis et al. (SODA '19). The third version uses the specific properties of the classical algorithm and treewidth, with a modified version of the quantum dynamic programming on the hypercube. Lastly, as a small side result, we also give a new classical time-space tradeoff for computing treewidth in $O^*(2^n)$ time and $O^*(\sqrt{2^n})$ space.
We initiate the study of parameterized complexity of $\textsf{QMA}$ problems in terms of the number of non-Clifford gates in the problem description. We show that for the problem of parameterized quantum circuit satisfiability, there exists a classical algorithm solving the problem with a runtime scaling exponentially in the number of non-Clifford gates but only polynomially with the system size. This result follows from our main result, that for any Clifford + $t$ $T$-gate quantum circuit satisfiability problem, the search space of optimal witnesses can be reduced to a stabilizer subspace isomorphic to at most $t$ qubits (independent of the system size). Furthermore, we derive new lower bounds on the $T$-count of circuit satisfiability instances and the $T$-count of the $W$-state assuming the classical exponential time hypothesis ($\textsf{ETH}$). Lastly, we explore the parameterized complexity of the quantum non-identity check problem.
In this paper, we provide a framework for constructing entanglement-assisted quantum error-correcting codes (EAQECCs) from classical additive codes over a finite commutative local Frobenius ring. We derive a formula for the minimum number of entanglement qudits required to construct an EAQECC from a linear code over the ring $\mathbb{Z}_{p^s}$. This is used to obtain an exact expression for the minimum number of entanglement qudits required to construct an EAQECC from an additive code over a Galois ring, which significantly extends known results for EAQECCs over finite fields.
Quantum teleportation allows one to transmit an arbitrary qubit from point A to point B using a pair of (pre-shared) entangled qubits and classical bits of information. The conventional protocol for teleportation uses two bits of classical information and assumes that the sender has access to only one copy of the arbitrary qubit to the sent. Here, we ask whether we can do better than two bits of classical information if the sender has access to multiple copies of the qubit to be teleported. We place no restrictions on the qubit states. Consequently, we propose a modified quantum teleportation protocol that allows Alice to reset the state of the entangled pair to its initial state using only local operations. As a result, the proposed teleportation protocol requires the transmission of only one classical bit with a probability greater than one-half. This has implications for efficient quantum communications and security of quantum cryptographic protocols based on quantum entanglement.
We introduce the hemicubic codes, a family of quantum codes obtained by associating qubits with the $p$-faces of the $n$-cube (for $n>p$) and stabilizer constraints with faces of dimension $(p\pm1)$. The quantum code obtained by identifying antipodal faces of the resulting complex encodes one logical qubit into $N = 2^{n-p-1} \tbinom{n}{p}$ physical qubits and displays local testability with a soundness of $\Omega(1/\log(N))$ beating the current state-of-the-art of $1/\log^{2}(N)$ due to Hastings. We exploit this local testability to devise an efficient decoding algorithm that corrects arbitrary errors of size less than the minimum distance, up to polylog factors. We then extend this code family by considering the quotient of the $n$-cube by arbitrary linear classical codes of length $n$. We establish the parameters of these generalized hemicubic codes. Interestingly, if the soundness of the hemicubic code could be shown to be constant, similarly to the ordinary $n$-cube, then the generalized hemicubic codes could yield quantum locally testable codes of length not exceeding an exponential or even polynomial function of the code dimension.
A fundamental computational problem is to find a shortest non-zero vector in Euclidean lattices, a problem known as the Shortest Vector Problem (SVP). This problem is believed to be hard even on quantum computers and thus plays a pivotal role in post-quantum cryptography. In this work we explore how (efficiently) Noisy Intermediate Scale Quantum (NISQ) devices may be used to solve SVP. Specifically, we map the problem to that of finding the ground state of a suitable Hamiltonian. In particular, (i) we establish new bounds for lattice enumeration, this allows us to obtain new bounds (resp.~estimates) for the number of qubits required per dimension for any lattices (resp.~random q-ary lattices) to solve SVP; (ii) we exclude the zero vector from the optimization space by proposing (a) a different classical optimisation loop or alternatively (b) a new mapping to the Hamiltonian. These improvements allow us to solve SVP in dimension up to 28 in a quantum emulation, significantly more than what was previously achieved, even for special cases. Finally, we extrapolate the size of NISQ devices that is required to be able to solve instances of lattices that are hard even for the best classical algorithms and find that with approximately $10^3$ noisy qubits such instances can be tackled.
In the classical world, the existence of commitments is equivalent to the existence of one-way functions. In the quantum setting, on the other hand, commitments are not known to imply one-way functions, but all known constructions of quantum commitments use at least one-way functions. Are one-way functions really necessary for commitments in the quantum world? In this work, we show that non-interactive quantum commitments (for classical messages) with computational hiding and statistical binding exist if pseudorandom quantum states exist. Pseudorandom quantum states are sets of quantum states that are efficiently generated but their polynomially many copies are computationally indistinguishable from the same number of copies of Haar random states [Ji, Liu, and Song, CRYPTO 2018]. It is known that pseudorandom quantum states exist even if $\BQP=\QMA$ (relative to a quantum oracle) [Kretschmer, TQC 2021], which means that pseudorandom quantum states can exist even if no quantum-secure classical cryptographic primitive exists. Our result therefore shows that quantum commitments can exist even if no quantum-secure classical cryptographic primitive exists. In particular, quantum commitments can exist even if no quantum-secure one-way function exists. In this work, we also consider digital signatures, which are other fundamental primitives in cryptography. We show that one-time secure digital signatures with quantum public keys exist if pseudorandom quantum states exist. In the classical setting, the existence of digital signatures is equivalent to the existence of one-way functions. Our result, on the other hand, shows that quantum signatures can exist even if no quantum-secure classical cryptographic primitive (including quantum-secure one-way functions) exists.
In 2017, Krenn reported that certain problems related to the perfect matchings and colourings of graphs emerge out of studying the constructability of general quantum states using modern photonic technologies. He realized that if we can prove that the \emph{weighted matching index} of a graph, a parameter defined in terms of perfect matchings and colourings of the graph is at most 2, that could lead to exciting insights on the potential of resources of quantum inference. Motivated by this, he conjectured that the {weighted matching index} of any graph is at most 2. The first result on this conjecture was by Bogdanov, who proved that the \emph{(unweighted) matching index} of graphs (non-isomorphic to $K_4$) is at most 2, thus classifying graphs non-isomorphic to $K_4$ into Type 0, Type 1 and Type 2. By definition, the weighted matching index of Type 0 graphs is 0. We give a structural characterization for Type 2 graphs, using which we settle Krenn's conjecture for Type 2 graphs. Using this characterization, we provide a simple $O(|V||E|)$ time algorithm to find the unweighted matching index of any graph. In view of our work, Krenn's conjecture remains to be proved only for Type 1 graphs. We give upper bounds for the weighted matching index in terms of connectivity parameters for such graphs. Using these bounds, for a slightly simplified version, we settle Krenn's conjecture for the class of graphs with vertex connectivity at most 2 and the class of graphs with maximum degree at most 4. Krenn has been publicizing his conjecture in various ways since 2017. He has even declared a reward for a resolution of his conjecture. We hope that this article will popularize the problem among computer scientists.
小貼士
登錄享
相關主題
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191