Whereas quantum complexity theory has traditionally been concerned with problems arising from classical complexity theory (such as computing boolean functions), it also makes sense to study the complexity of inherently quantum operations such as constructing quantum states or performing unitary transformations. With this motivation, we define models of interactive proofs for synthesizing quantum states and unitaries, where a polynomial-time quantum verifier interacts with an untrusted quantum prover, and a verifier who accepts also outputs an approximation of the target state (for the state synthesis problem) or the result of the target unitary applied to the input state (for the unitary synthesis problem); furthermore there should exist an "honest" prover which the verifier accepts with probability 1. Our main result is a "state synthesis" analogue of the inclusion PSPACE $\subseteq$ IP: any sequence of states computable by a polynomial-space quantum algorithm (which may run for exponential time) admits an interactive protocol of the form described above. Leveraging this state synthesis protocol, we also give a unitary synthesis protocol for polynomial space-computable unitaries that act nontrivially on only a polynomial-dimensional subspace.
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It is well known that an arbitrary $n$-qubit quantum state $|\Psi\rangle$ can be prepared with $\Theta(2^n)$ two-qubit gates. In this work, we investigate the task in a "straddling gates" scenario: consider $n$ qubits divided equally into two sets and gates within each set are free; what is the least cost of two-qubit gates straddling the sets (also known as the "binding complexity") for preparing an arbitrary quantum state, assuming no ancilla qubits allowed? In this work, we give an algorithm that fulfills the task with $O(n^2 2^{n/2})$ straddling gates, which nearly matches the lower bound to a lower order factor. We then prove any $U(2^n)$ decomposition requires no more than $O(2^{n})$ straddling gates. This resolves an open problem posed by Vijay Balasubramanian, who was motivated by the "Complexity=Volume" conjecture in AdS/CFT theory. Furthermore, we extend our discussion to multi-partite systems, define a novel binding complexity class, the "Schmidt decomposable" states, and give a circuit construction explanation for its unique property. Lastly, we reveal binding complexity's significance, comparing it to Von Neumann entropy as an entanglement measure.
Probabilistic programming languages aid developers performing Bayesian inference. These languages provide programming constructs and tools for probabilistic modeling and automated inference. Prior work introduced a probabilistic programming language, ProbZelus, to extend probabilistic programming functionality to unbounded streams of data. This work demonstrated that the delayed sampling inference algorithm could be extended to work in a streaming context. ProbZelus showed that while delayed sampling could be effectively deployed on some programs, depending on the probabilistic model under consideration, delayed sampling is not guaranteed to use a bounded amount of memory over the course of the execution of the program. In this paper, we the present conditions on a probabilistic program's execution under which delayed sampling will execute in bounded memory. The two conditions are dataflow properties of the core operations of delayed sampling: the $m$-consumed property and the unseparated paths property. A program executes in bounded memory under delayed sampling if, and only if, it satisfies the $m$-consumed and unseparated paths properties. We propose a static analysis that abstracts over these properties to soundly ensure that any program that passes the analysis satisfies these properties, and thus executes in bounded memory under delayed sampling.
The number of down-steps between pairs of up-steps in $k_t$-Dyck paths, a generalization of Dyck paths consisting of steps $\{(1, k), (1, -1)\}$ such that the path stays (weakly) above the line $y=-t$, is studied. Results are proved bijectively and by means of generating functions, and lead to several interesting identities as well as links to other combinatorial structures. In particular, there is a connection between $k_t$-Dyck paths and perforation patterns for punctured convolutional codes (binary matrices) used in coding theory. Surprisingly, upon restriction to usual Dyck paths this yields a new combinatorial interpretation of Catalan numbers.
We investigate variational principles for the approximation of quantum dynamics that apply for approximation manifolds that do not have complex linear tangent spaces. The first one, dating back to McLachlan (1964) minimizes the residuum of the time-dependent Schr\"odinger equation, while the second one, originating from the lecture notes of Kramer--Saraceno (1981), imposes the stationarity of an action functional. We characterize both principles in terms of metric and a symplectic orthogonality conditions, consider their conservation properties, and derive an elementary a-posteriori error estimate. As an application, we revisit the time-dependent Hartree approximation and frozen Gaussian wave packets.
Steganography is the science of hiding and communicating a secret message by embedding it in an innocent looking text such that the eavesdropper is unaware of its existence. Previously, attempts were made to establish steganography using quantum key distribution (QKD). Recently, it has been shown that such protocols are vulnerable to a certain steganalysis attack that can detect the presence of the hidden message and suppress the entire communication. In this work, we elaborate on the vulnerabilities of the original protocol which make it insecure against this detection attack. Further, we propose a novel steganography protocol using discrete modulation continuous variable QKD that eliminates the threat of this detection-based attack. Deriving from the properties of our protocol, we also propose modifications in the original protocol to dispose of its vulnerabilities and make it insusceptible to steganalysis.
Continuous-time quantum walks have proven to be an extremely useful framework for the design of several quantum algorithms. Often, the running time of quantum algorithms in this framework is characterized by the quantum hitting time: the time required by the quantum walk to find a vertex of interest with a high probability. In this article, we provide improved upper bounds for the quantum hitting time that can be applied to several CTQW-based quantum algorithms. In particular, we apply our techniques to the glued-trees problem, improving their hitting time upper bound by a polynomial factor: from $O(n^5)$ to $O(n^2\log n)$. Furthermore, our methods also help to exponentially improve the dependence on precision of the continuous-time quantum walk based algorithm to find a marked node on any ergodic, reversible Markov chain by Chakraborty et al. [PRA 102, 022227 (2020)].
We propose a novel approach to program synthesis, focusing on synthesizing database queries. At a high level, our proposed algorithm takes as input a sketch with soft constraints encoding user intent, and then iteratively interacts with the user to refine the sketch. At each step, our algorithm proposes a candidate refinement of the sketch, which the user can either accept or reject. By leveraging this rich form of user feedback, our algorithm is able to both resolve ambiguity in user intent and improve scalability. In particular, assuming the user provides accurate inputs and responses, then our algorithm is guaranteed to converge to the true program (i.e., one that the user approves) in polynomial time. We perform a qualitative evaluation of our algorithm, showing how it can be used to synthesize a variety of queries on a database of academic publications.
Greedy algorithms have long been a workhorse for learning graphical models, and more broadly for learning statistical models with sparse structure. In the context of learning directed acyclic graphs, greedy algorithms are popular despite their worst-case exponential runtime. In practice, however, they are very efficient. We provide new insight into this phenomenon by studying a general greedy score-based algorithm for learning DAGs. Unlike edge-greedy algorithms such as the popular GES and hill-climbing algorithms, our approach is vertex-greedy and requires at most a polynomial number of score evaluations. We then show how recent polynomial-time algorithms for learning DAG models are a special case of this algorithm, thereby illustrating how these order-based algorithms can be rigourously interpreted as score-based algorithms. This observation suggests new score functions and optimality conditions based on the duality between Bregman divergences and exponential families, which we explore in detail. Explicit sample and computational complexity bounds are derived. Finally, we provide extensive experiments suggesting that this algorithm indeed optimizes the score in a variety of settings.
In recent decades, with the emergence of numerous novel intelligent optimization algorithms, many optimization researchers have begun to look for a basic search mechanism for their schemes that provides a more essential explanation of their studies. This paper aims to study the basic mechanism of an algorithm for black-box optimization with quantum theory. To achieve this goal, the Schroedinger equation is employed to establish the relationship between the optimization problem and the quantum system, which makes it possible to study the dynamic search behaviors in the evolution process with quantum theory. Moreover, to explore the basic behavior of the optimization system, the optimization problem is assumed to be decomposed and approximated. Then, a multilevel approximation quantum dynamics model of the optimization algorithm is established, which provides a mathematical and physical framework for the analysis of the optimization algorithm. Correspondingly, the basic search behavior based on this model is derived, which is governed by quantum theory. Comparison experiments and analysis between different bare-bones algorithms confirm the existence of the quantum mechanic based basic search mechanism of the algorithm on black-box optimization.
Quantum switches are critical components in quantum networks, distributing maximally entangled pairs among end nodes by entanglement swapping. In this work, we design protocols that schedule entanglement swapping operations in quantum switches. Entanglement requests randomly arrive at the switch, and the goal of an entanglement swapping protocol is to stabilize the quantum switch so that the number of unfinished entanglement requests is bounded with a high probability. We determine the capacity region for the rates of entanglement requests and develop entanglement swapping protocols to stabilize the switch. Among these protocols, the on-demand protocols are not only computationally efficient, but also achieve high fidelity and low latency demonstrated by results obtained using a quantum network discrete event simulator.
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