Given a local Hamiltonian, how difficult is it to determine the entanglement structure of its ground state? We show that this problem is computationally intractable even if one is only trying to decide if the ground state is volume-law vs near area-law entangled. We prove this by constructing strong forms of pseudoentanglement in a public-key setting, where the circuits used to prepare the states are public knowledge. In particular, we construct two families of quantum circuits which produce volume-law vs near area-law entangled states, but nonetheless the classical descriptions of the circuits are indistinguishable under the Learning with Errors (LWE) assumption. Indistinguishability of the circuits then allows us to translate our construction to Hamiltonians. Our work opens new directions in Hamiltonian complexity, for example whether it is difficult to learn certain phases of matter.

Entanglement is a quantum resource, in some ways analogous to randomness in classical computation. Inspired by recent work of Gheorghiu and Hoban, we define the notion of "pseudoentanglement'', a property exhibited by ensembles of efficiently constructible quantum states which are indistinguishable from quantum states with maximal entanglement. Our construction relies on the notion of quantum pseudorandom states -- first defined by Ji, Liu and Song -- which are efficiently constructible states indistinguishable from (maximally entangled) Haar-random states. Specifically, we give a construction of pseudoentangled states with entanglement entropy arbitrarily close to $\log n$ across every cut, a tight bound providing an exponential separation between computational vs information theoretic quantum pseudorandomness. We discuss applications of this result to Matrix Product State testing, entanglement distillation, and the complexity of the AdS/CFT correspondence. As compared with a previous version of this manuscript (arXiv:2211.00747v1) this version introduces a new pseudorandom state construction, has a simpler proof of correctness, and achieves a technically stronger result of low entanglement across all cuts simultaneously.

We give a polynomial time classical algorithm for sampling from the output distribution of a noisy random quantum circuit in the regime of anti-concentration to within inverse polynomial total variation distance. The algorithm is based on a quantum analog of noise induced low degree approximations of Boolean functions, which takes the form of the truncation of a Feynman path integral in the Pauli basis.

(Sender-)Deniable encryption provides a very strong privacy guarantee: a sender who is coerced by an attacker into “opening” their ciphertext after-the-fact is able to generate “fake” local random choices that are consistent with any plaintext of their choice. The only known fully-efficient constructions of public-key deniable encryption rely on indistinguishability obfuscation (iO) (which currently can only be based on sub-exponential hardness assumptions).

We demonstrate the possibility of (sub)exponential quantum speedup via a quantum algorithm that follows an adiabatic path of a gapped Hamiltonian with no sign problem. The Hamiltonian that exhibits this speed-up comes from the adjacency matrix of an undirected graph whose vertices are labeled by n-bit strings, and we can view the adiabatic evolution as an efficient O(poly(n))-time quantum algorithm for finding a specific “EXIT” vertex in the graph given the “ENTRANCE” vertex. On the other hand we show that if the graph is given via an adjacency-list oracle, there is no classical algorithm that finds the “EXIT” with probability greater than exp(−nδ) using at most exp(nδ) queries for δ= 1/5 − o(1). Our construction of the graph is somewhat similar to the “welded-trees” construction of Childs et al., but uses additional ideas of Hastings for achieving a spectral gap and a short adiabatic path.

We consider a new model for the testing of untrusted quantum devices, consisting of a single polynomial time bounded quantum device interacting with a classical polynomial time verifier. In this model, we propose solutions to two tasks—a protocol for efficient classical verification that the untrusted device is “truly quantum” and a protocol for producing certifiable randomness from a single untrusted quantum device. Our solution relies on the existence of a new cryptographic primitive for constraining the power of an untrusted quantum device: post-quantum secure trapdoor claw-free functions that must satisfy an adaptive hardcore bit property. We show how to construct this primitive based on the hardness of the learning with errors (LWE) problem.

The AdS/CFT correspondence is central to efforts to reconcile gravity and quantum mechanics, a fundamental goal of physics. It posits a duality between a gravitational theory in Anti de Sitter (AdS) space and a quantum mechanical conformal field theory (CFT), embodied in a map known as the AdS/CFT dictionary mapping states to states and operators to operators. This dictionary map is not well understood and has only been computed on special, structured instances. In this work we introduce cryptographic ideas to the study of AdS/CFT, and provide evidence that either the dictionary must be exponentially hard to compute, or else the quantum Extended Church-Turing thesis must be false in quantum gravity. Our argument has its origins in a fundamental paradox in the AdS/CFT correspondence known as the wormhole growth paradox. The paradox is that the CFT is believed to be "scrambling" - i.e. the expectation value of local operators equilibrates in polynomial time - whereas the gravity theory is not, because the interiors of certain black holes known as "wormholes" do not equilibrate and instead their volume grows at a linear rate for at least an exponential amount of time. So what could be the CFT dual to wormhole volume? Susskind’s proposed resolution was to equate the wormhole volume with the quantum circuit complexity of the CFT state. From a computer science perspective, circuit complexity seems like an unusual choice because it should be difficult to compute, in contrast to physical quantities such as wormhole volume. We show how to create pseudorandom quantum states in the CFT, thereby arguing that their quantum circuit complexity is not "feelable", in the sense that it cannot be approximated by any efficient experiment. This requires a specialized construction inspired by symmetric block ciphers such as DES and AES, since unfortunately existing constructions based on quantum-resistant one way functions cannot be used in the context of the wormhole growth paradox as only very restricted operations are allowed in the CFT. By contrast we argue that the wormhole volume is "feelable" in some general but non-physical sense. The duality between a "feelable" quantity and an "unfeelable" quantity implies that some aspect of this duality must have exponential complexity. More precisely, it implies that either the dictionary is exponentially complex, or else the quantum gravity theory is exponentially difficult to simulate on a quantum computer. While at first sight this might seem to justify the discomfort of complexity theorists with equating computational complexity with a physical quantity, a further examination of our arguments shows that any resolution of the wormhole growth paradox must equate wormhole volume to an "unfeelable" quantity, leading to the same conclusions. In other words this discomfort is an inevitable consequence of the paradox.

A critical milestone on the path to useful quantum computers is quantum supremacy - a demonstration of a quantum computation that is prohibitively hard for classical computers. A leading near-term candidate, put forth by the Google/UCSB team, is sampling from the probability distributions of randomly chosen quantum circuits, which we call Random Circuit Sampling (RCS). In this paper we study both the hardness and verification of RCS. While RCS was defined with experimental realization in mind, we show complexity theoretic evidence of hardness that is on par with the strongest theoretical proposals for supremacy. Specifically, we show that RCS satisfies an average-case hardness condition - computing output probabilities of typical quantum circuits is as hard as computing them in the worst-case, and therefore #P-hard. Our reduction exploits the polynomial structure in the output amplitudes of random quantum circuits, enabled by the Feynman path integral. In addition, it follows from known results that RCS satisfies an anti-concentration property, making it the first supremacy proposal with both average-case hardness and anti-concentration.

We give a protocol for producing certifiable randomness from a single untrusted quantum device that is polynomial-time bounded. The randomness is certified to be statistically close to uniform from the point of view of any computationally unbounded quantum adversary, that may share entanglement with the quantum device. The protocol relies on the existence of post-quantum secure trapdoor claw-free functions, and introduces a new primitive for constraining the power of an untrusted quantum device. We then show how to construct this primitive based on the hardness of the learning with errors (LWE) problem. The randomness protocol can also be used as the basis for an efficiently verifiable "quantum supremacy" proposal, thus answering an outstanding challenge in the field.

The No Low-Energy Trivial States (NLTS) conjecture of Freedman and Hastings (Quantum Information and Computation 2014), which asserts the existence of local Hamiltonians whose low-energy states cannot be generated by constant-depth quantum circuits, identifies a fundamental obstacle to resolving the quantum PCP conjecture. Progress towards the NLTS conjecture was made by Eldar and Harrow (Foundations of Computer Science 2017), who proved a closely related theorem called No Low-Error Trivial States (NLETS). In this paper, we give a much simpler proof of the NLETS theorem and use the same technique to establish superpolynomial circuit size lower bounds for noisy ground states of local Hamiltonians (assuming QCMA != QMA), resolving an open question of Eldar and Harrow. We discuss the new light our results cast on the relationship between NLTS and NLETS. Finally, our techniques imply the existence of approximate quantum low-weight check (qLWC) codes with linear rate, linear distance, and constant weight checks. These codes are similar to quantum LDPC codes except (1) each particle may participate in a large number of checks, and (2) errors only need to be corrected up to fidelity 1 - 1/poly(n). This stands in contrast to the best-known stabilizer LDPC codes due to Freedman, Meyer, and Luo which achieve a distance of O(sqrt{n log n}). The principal technique used in our results is to leverage the Feynman-Kitaev clock construction to approximately embed a subspace of states defined by a circuit as the ground space of a local Hamiltonian.

One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance by Landau et al. gave a polynomial time algorithm to compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unresolved, including whether there exist rigorous efficient algorithms when the ground space is degenerate (and poly(n) dimensional), or for the poly(n) lowest energy states for 1D systems, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm for finding low energy states for 1D systems, based on a rigorously justified renormalization group (RG)-type transformation. In the process we resolve some of the aforementioned open questions, including giving a polynomial time algorithm for poly(n) degenerate ground spaces and an n^(O(log n)) algorithm for the poly(n) lowest energy states for 1D systems (under a mild density condition). We note that for these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time is O(nM(n)), where M(n) is the time required to multiply two n by n matrices.

We consider multiplayer games in which the players fall in two teams of size k, with payoffs equal within, and of opposite sign across, the two teams. In the classical case of k=1, such zero-sum games possess a unique value, independent of order of play, due to the von Neumann minimax theorem. However, this fails for all k>1; we can measure this failure by a duality gap, which quantifies the benefit of being the team to commit last to its strategy. In our main result we show that the gap equals 2(1-2^{1-k}) for m=2 and 2(1-\m^{-(1-o(1))k}) for m>2, with m being the size of the action space of each player. At a finer level, the cost to a team of individual players acting independently while the opposition employs joint randomness is 1-2^{1-k} for k=2, and 1-\m^{-(1-o(1))k} for m>2. This class of multiplayer games, apart from being a natural bridge between two-player zero-sum games and general multiplayer games, is motivated from Biology (the weak selection model of evolution) and Economics (players with shared utility but poor coordination).

We introduce a technique for applying quantum expanders in a distributed fashion, and use it to solve two basic questions: testing whether a bipartite quantum state shared by two parties is the maximally entangled state and disproving a generalized area law. In the process these two questions which appear completely unrelated turn out to be two sides of the same coin. Strikingly in both cases a constant amount of resources are used to verify a global property.

Even the most seasoned students of evolution, starting with Darwin himself, have occasionally expressed amazement at the fact that the mechanism of natural selection has produced the whole of Life as we see it around us. From a computational perspective, it is natural to marvel at evolution's solution to the problems of robotics, vision and theorem proving! What, then, is the complexity of evolution, viewed as an algorithm? One answer to this question is 10^{12}, roughly the number of sequential steps or generations from the earliest single celled creatures to today's Homo Sapiens. To put this into perspective, the processor of a modern cell phone can perform 10^{12} steps in less than an hour. Another answer is 10^30, the degree of parallelism, roughly the maximum number of organisms living on the Earth at any time. Perhaps the answer should be the product of the two numbers, roughly 10^42, to reflect the total work done by evolution, viewed as a parallel algorithm. Here we argue, interpreting our recently published paper, that none of the above answers is really correct. Viewing evolution as an algorithm poses an additional challenge: recombination. Even if evolution succeeds in producing a particularly good solution (a highly fit individual), its offspring would only inherit half its genes, and therefore appear unlikely to be a good solution. This is the core of the problem of explaining the role of sex in evolution, known as the "queen of problems in evolutionary biology". The starting point is the diffusion-equation-based approach of theoretical population geneticists, who analyze the changing allele frequencies (over the generations) in the gene pool, consisting of the aggregate of the genetic variants (or "alleles") over all genes (or "loci") and over all individuals in a species. Taking this viewpoint to its logical conclusion, rather than acting on individuals or species or genes, evolution acts on this gene pool, or genetic soup, by making it more "potent", in the sense that it increases the expected fitness of genotype drawn randomly from this soup. Moreover, for much genetic variation, this soup may be assumed to be in the regime of weak selection, a regime where the probability of occurrence of a certain genotype involving various alleles at different loci is simply the product of the probabilities of each of its alleles. In this regime, we show that evolution in the regime of weak selection can be formulated as a game, where the recombining loci are the players, the alleles in those loci are possible moves or actions of each player, and the expected payoff of each player-locus is precisely the organism's expected fitness across the genotypes that are present in the population. Moreover, the dynamics specified by the diffusion equations of theoretical population geneticists is closely approximated by the dynamics of multiplicative weight updates (MWUA). The algorithmic connection to MWUA brings with it new insights for evolutionary biology, specifically, into the question of how genetic diversity is maintained in the presence of natural selection. For this it is useful to consider a dual view of MWUA, which expresses "what each gene is optimizing" as it plays the game. Remarkably this turns out to be a particular convex combination of the entropy of its distribution over alleles and cumulative expected fitness. This sheds new light on the maintenance of diversity in evolution. All of this suggests that the complexity of evolution should indeed be viewed as 10^12, but for a subtle reason. It is the number of steps of multiplicative weight updates carried out on allele frequencies in the genetic soup. A closer examination of this reveals further that the accurate tracking of allele frequencies over the generations requires the simulation of a quadratic dynamical system (two parents for each offspring). Moreover the simulation of even simple quadratic dynamical systems is known to be PSPACE-hard. This suggests that the tracking of allele frequencies might require large population sizes for each species, putting into perspective the number 10^30. Finally, it is worth noting that in this view there is a primacy to recombination or sex, which serve to provide robustness to the mechanism of evolution, as well as the framework within which MWUA operates.

Computing ground states of local Hamiltonians is a fundamental problem in condensed matter physics. The problem is known to be QMA-complete, even for one-dimensional Hamiltonians [1]. This means that we do not even expect that there is a sub-exponential size description of the ground state that allows efficient computation of local observables such as the energy. In sharp contrast, the heuristic density matrix renormalization group (DMRG) algorithm invented two decades ago [5] has been remarkably successful in practice on one-dimensional problems. The situation is reminiscent of the unexplained success of the simplex algorithm before the advent of ellipsoid and interior-point methods. Is there a principled explanation for this, in the form of a large class of one-dimensional Hamiltonians whose ground states can be provably efficiently approximated? Here we give such an algorithm for gapped one-dimensional Hamiltonians: our algorithm outputs an (inverse-polynomial) approximation to the ground state, expressed as a matrix product state (MPS) of polynomial bond dimension. The running time of the algorithm is polynomial in the number of qudits n and the approximation quality δ, for a fixed local dimension d and gap Δ > 0.

Quantum cryptography is based on the discovery that the laws of quantum mechanics allow levels of security that are impossible to replicate in a classical world [2, 8, 12]. Can such levels of security be guaranteed even when the quantum devices on which the protocol relies are untrusted? This fundamental question in quantum cryptography dates back to the early nineties when the challenge of achieving device independent quantum key distribution, or DIQKD, was first formulated [9]. We answer this challenge affirmatively by exhibiting a robust protocol for DIQKD and rigorously proving its security. The protocol achieves a linear key rate while tolerating a constant noise rate in the devices. The security proof assumes only that the devices can be modeled by the laws of quantum mechanics and are spatially isolated from each other and any adversary's laboratory. In particular, we emphasize that the devices may have quantum memory. All previous proofs of security relied either on the use of many independent pairs of devices [6, 4, 7], or on the absence of noise [10, 1].

In this paper we point out a new and unexpected connection between three fields: Evolution Theory, Game Theory, and Algorithms.

Can a classical experimentalist command an untrusted quantum system to realize arbitrary quantum dynamics, aborting if it misbehaves? If so, then we could realize the dream of device-independent quantum cryptography: using untrusted quantum devices to establish a shared random key, with security based on the correctness of quantum mechanics. It would also allow for testing whether a claimed quantum computer is truly quantum. We prove a rigidity theorem for the famous Clauser-Horne-Shimony-Holt (CHSH) game, first formulated to provide a means of experimentally testing the violation of the Bell inequalities. The theorem shows that the only way for the two non-communicating quantum players to win many games played in sequence is if their shared quantum state is close to the tensor product of EPR states (Bell states) and their measurements are the optimal CHSH measurements on successive qubits. This theorem may be viewed as analogous to classical multi-linearity testing, in the sense that the outcome of local checks gives a characterization of a global object.

We introduce a protocol through which a pair of quantum mechanical devices may be used to generate n bits that are ε-close in statistical distance from n uniformly distributed bits, starting from a seed of O(log n log 1/ε) uniform bits. The bits generated are certifiably random based only on a simple statistical test that can be performed by the user, and on the assumption that the devices do not communicate in the middle of each phase of the protocol. No other assumptions are placed on the devices' inner workings. A modified protocol uses a seed of O(log3 n) uniformly random bits to generate n bits that are poly-1(n)-indistinguishable from uniform even from the point of view of a quantum adversary who may have had prior access to the devices, and may be entangled with them.

The classical description of quantum states is in general exponential in the number of qubits. Can we get polynomial descriptions for more restricted sets of states such as ground states of interesting subclasses of local Hamiltonians? This is the basic problem in the study of the complexity of ground states, and requires an understanding of multi-particle entanglement and quantum correlations in such states. Area laws provide a fundamental ingredient in the study of the complexity of ground states, since they offer a way to bound in a quantitative way the entanglement in such states. Although they have long been conjectured for many body systems in arbitrary dimensions, a general rigorous was only recently proved in Hastings' seminal paper [8] for ID systems. In this paper, we give a combinatorial proof of the ID area law for the special case of frustration free systems, improving by an exponential factor the scaling in terms of the inverse spectral gap and the dimensionality of the particles. The scaling in terms of the dimension of the particles is a potentially important issue in the context of resolving the 2D case and higher dimensions, which is one of the most important open questions in Hamiltonian complexity. Our proof is based on a reformulation of the detectability lemma, introduced by us in the context of quantum gap amplification [1]. We give an alternative proof of the detectability lemma, which is not only simpler and more intuitive than the original proof, but also removes a key restriction in the original statement, making it more suitable for this new context. We also give a one page proof of Hastings' proof that the correlations in the ground states of gapped Hamiltonians decay exponentially with the distance, demonstrating the simplicity of the combinatorial approach for those problems.

Quantum states generally require exponential sized classical descriptions, but the long conjectured area law provides hope that a large class of natural quantum states can be described succinctly. Recent progress in formally proving the area law is described.

The quantum analog of a constraint satisfaction problem is a sum of local Hamiltonians- each (term of the) Hamiltonian specifies a local constraint whose violation contributes to the energy of the given quantum state. Formalizing the intuitive connection between the ground (minimal) energy of the Hamiltonian and the minimum number of violated constraints is problematic, since the number of constraints being violated is not well defined when the terms in the Hamiltonian do not commute. The detectability lemma proved in this paper provides precisely such a quantitative connection. We apply the lemma to derive a quantum analogue of the classical gap amplification lemma of random walks on expander graphs. The quantum gap amplification lemma holds for local Hamiltonians with expander interaction graphs. Our proofs are based on a novel structure imposed on the Hilbert space, which we call the XY decomposition, which enables a reduction from the quantum non-commuting case to the commuting case (where many classical arguments go through). The results may have several interesting implications. First, proving a quantum analogue to the PCP theorem is one of the most important challenges in quantum complexity theory. Our quantum

We give aO(√logn)-approximation algorithm for the sparsest cut, edge expansion, balanced separator, and graph conductance problems. This improves theO(logn)-approximation of Leighton and Rao (1988). We use a well-known semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets inRd, whose proof makes essential use of a phenomenon called measure concentration. We also describe an interesting and natural “approximate certificate” for a graph's expansion, which involves embedding ann-node expander in it with appropriate dilation and congestion. We call this an expander flow.

We show that the sparsest cut in graphs with n vertices and m edges can be approximated within O (log 2 n ) factor in Õ( m + n 3/2 ) time using polylogarithmic single commodity max-flow computations. Previous algorithms are based on multicommodity flows that take time Õ( m + n 2 ). Our algorithm iteratively employs max-flow computations to embed an expander flow, thus providing a certificate of expansion. Our technique can also be extended to yield an O (log 2 n )-(pseudo-) approximation algorithm for the edge-separator problem with a similar running time.

In this paper we obtain improved upper and lower bounds for the best approximation factor for Sparsest Cut achievable in the cut-matching game framework proposed in Khandekar et al. [9]. We show that this simple framework can be used to design combinatorial algorithms that achieve O(log n) approximation factor and whose running time is dominated by a poly-logarithmic number of single-commodity max-flow computations. This matches the performance of the algorithm of Arora and Kale [2]. Moreover, we also show that it is impossible to get an approximation factor of better than Ω(√log n) in the cut-matching game framework. These results suggest that the simple and concrete abstraction of the cut-matching game may be powerful enough to capture the essential features of the complexity of Sparsest Cut.

Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonAbelian hidden subgroup problem, which generalizes the central problem solved by Shor's factoring algorithm. We suggest an alternative generalization, namely to problems of finding hidden nonlinear structures over finite fields. We give examples of two such problems that can be solved efficiently by a quantum computer, but not by a classical computer. We also give some positive results on the quantum query complexity of finding hidden nonlinear structures.

Quantum physics is a fascinating area from a computational viewpoint. The features that make quantum systems prohibitively hard to simulate classically are precisely the aspects exploited by quantum computation to obtain exponential speedups over classical computers. In this talk I will survey our current understanding of the power (and limits) of quantum computers, and prospects for experimentally realizing them in the near future. I will also touch upon insights from quantum comuptation that have resulted in new classical algorithms for efficient simulation of certain important quantum systems.

How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a trade-off revealing LP and use it to derive an optimal algorithm achieving a competitive ratio of 1−1/ e for this problem.

We show that the sparsest cut in graphs can be approximated within O(log2 n) factor in Õ(n3/2) time using polylogarithmic single commodity max-flow computations. Previous algorithms are based on multicommodity flows which take time Õ(n2). Our algorithm iteratively employs max-flow computations to embed an expander flow, thus providing a certificate of expansion. Our technique can also be extended to yield an O(log2 n) (pseudo) approximation algorithm for the edge-separator problem with a similar running time.

Device initialization is a difficult challenge in some proposed realizations of quantum computers, and as such, must be treated as a computational resource. The degree of initialization can be quantified by k , the number of clean qubits in the initial state of the register. In this article, we show that unless m ∈ O ( k + log n ), oblivious (gate-by-gate) simulation of an ideal m -qubit quantum circuit by an n -qubit circuit with k clean qubits is impossible. Effectively, this indicates that there is no avoiding physical initialization of a quantity of qubits proportional to that required by the best ideal quantum circuit.

How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a tradeoff revealing LP and use it to derive two optimal algorithms achieving competitive ratios of 1-1/e for this problem.

We give a O(√log n)-approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)-approximation of Leighton and Rao (1988). We use a well-known semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in Rd, whose proof makes essential use of a phenomenon called measure concentration. We also describe an interesting and natural "certificate" for a graph's expansion, by embedding an n-node expander in it with appropriate dilation and congestion. We call this an expander flow.

Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function $f: X \times Y \rightarrow \{0,1\}$ and a probability distribution ${\cal D}$ over $X \times Y$, we define the sampling complexity of $(f, {\cal D})$ as the minimum number of bits that Alice and Bob must communicate for Alice to pick $x \in X$ and Bob to pick $y \in Y$ as well as a value z such that the resulting distribution of $(x,y,z)$ is close to the distribution $({\cal D}, f({\cal D}))$. In this paper we initiate the study of sampling complexity, in both the classical and quantum models. We give several variants of a definition. We completely characterize some of these variants and give upper and lower bounds on others. In particular, this allows us to establish an exponential gap between quantum and classical sampling complexity for the set-disjointness function.

We consider the possibility of encoding m classical bits into many fewer n quantum bits (qubits) so that an arbitrary bit from the original m bits can be recovered with good probability. We show that nontrivial quantum codes exist that have no classical counterparts. On the other hand, we show that quantum encoding cannot save more than a logarithmic additive factor over the best classical encoding. The proof is based on an entropy coalescence principle that is obtained by viewing Holevo's theorem from a new perspective.In the existing implementations of quantum computing, qubits are a very expensive resource. Moreover, it is difficult to reinitialize existing bits during the computation. In particular, reinitialization is impossible in NMR quantum computing, which is perhaps the most advanced implementation of quantum computing at the moment. This motivates the study of quantum computation with restricted memory and no reinitialization, that is, of quantum finite automata. It was known that there are languages that are recognized by quantum finite automata with sizes exponentially smaller than those of corresponding classical automata. Here, we apply our technique to show the surprising result that there are languages for which quantum finite automata take exponentially more states than those of corresponding classical automata.

The authors analyze the computational power and limitations of the recently proposed 'quantum adiabatic evolution algorithm'. Adiabatic quantum computation is a novel paradigm for the design of quantum algorithms; it is truly quantum in the sense that it can be used to speed up searching by a quadratic factor over any classical algorithm. On the question of whether this new paradigm may be used to efficiently solve NP-complete problems on a quantum computer, we show that the usual query complexity arguments cannot be used to rule out a polynomial time solution. On the other hand, we argue that the adiabatic approach may be thought of as a kind of 'quantum local search'. We design a family of minimization problems that is hard for such local search heuristics, and establish an exponential lower bound for the adiabatic algorithm for these problems. This provides insights into the limitations of this approach. It remains an open question whether adiabatic quantum computation can establish an exponential speed-up over traditional computing or if there exists a classical algorithm that can simulate the quantum adiabatic process efficiently.

We set the ground for a theory of quantum walks on graphs-the generalization of random walks on finite graphs to the quantum world. Such quantum walks do not converge to any stationary distribution, as they are unitary and reversible. However, by suitably relaxing the definition, we can obtain a measure of how fast the quantum walk spreads or how confined the quantum walk stays in a small neighborhood. We give definitions of mixing time, filling time, dispersion time. We show that in all these measures, the quantum walk on the cycle is almost quadratically faster then its classical correspondent. On the other hand, we give a lower bound on the possible speed up by quantum walks for general graphs, showing that quantum walks can be at most polynomially faster than their classical counterparts.

We provide positive and negative results concerning the “standard method” of identifying a hidden subgroup of a nonabelian group using a quantum computer.

There is strong evidence that computers based upon the principles of quantum physics represent an inherently new and more powerful model of computation. Such computers violate the modern form of the Church-Turing thesis (which lies at the foundations of computer science). This thesis can be informally summarized as follows: all physical implementations of computing devices can be described by the same abstract model of computation-the probabilistic Turing Machine or the equivalent random access machine. In this paper, we will describe two formal models for quantum computers: quantum circuits and quantum Turing Machines, introduced by Deutsch [1985].

Article Free Access Share on Quantum bit escrow Authors: Dorit Aharonov University of California, Berkeley, CA University of California, Berkeley, CAView Profile , Amnon Ta-Shma University of California, Berkeley, CA University of California, Berkeley, CAView Profile , Umesh V. Vazirani University of California, Berkeley, CA University of California, Berkeley, CAView Profile , Andrew C. Yao Princeton University, Princeton, NJ Princeton University, Princeton, NJView Profile Authors Info & Claims STOC '00: Proceedings of the thirty-second annual ACM symposium on Theory of computingMay 2000 Pages 705–714https://doi.org/10.1145/335305.335404Published:01 May 2000Publication History 46citation619DownloadsMetricsTotal Citations46Total Downloads619Last 12 Months56Last 6 weeks6 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

We consider quantum computing in the one-qubit model where the starting state of a quantum computer consists of k qubits in a pure state and n − k qubits in a maximally mixed state. We ask the following question: is there a general method for simulating an arbitrary m-qubit pure state quantum computation by a quantum computation in the k-qubit model? We show that, under certain constraints, this is impossible, unless m = O(k + log n). 1.

We consider the possibility of encoding m classical bits into much fewer n quantum bits so that an arbitrary bit from the original m bits can be recovered with a good probability, and we show that non-trivial quantum encodings exist that have no classical counterparts.On the other hand, we show that quantum encodings cannot be much more succint as compared to classical encodings, and we provide a lower bound on such quantum encodings.Finally, using this lower bound, we prove an exponential lower bound an the size of l-way quantum linite automata for a family of languages accepted by linear sized deterministic linite automata.

We describe a quantum mechanical heat engine.Like its classical counterpart introduced by Carnot, this entine carries out a reversible process in which an input of energy to the system results in a separation of cold and hot regions.The method begins with a reinterpretation in thermodynamic terms of a simple step introduced by van Neumann to extract fair coin flips from sequences of biased coin flips.Some of the experimental set-ups proposed for implementation of quantum computers, begin with the quantum bits of the computer initially in a mixed state.Each qubit is L polarized -in the state IO) with probability 9, and in the state 11) with probability *, independently (or nearly so) of all other bits.The heat engine may be used to trans.form this initial collection of n qubits into a state in which a near-optimal m = n[ FIg(l +e) + %Ig(l -c) -o(l)] qubits are in the joint state IO"').These qubits can then be used as the register for a quantum computation.The heat engine is described at the level of an algorithm implementable in any quantum system capable of massive coherent states.A particular implementation is also described for a system of nuclear spins arranged in a chain.The temperature the cold qubits reach is inverse polynomial in n.

Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function f: X/spl times/Y/spl rarr/{0,1} and a probability distribution D over X/spl times/Y, we define the sampling complexity of (f,D) as the minimum number of bits Alice and Bob must communicate for Alice to pick x/spl isin/X and Bob to pick y/spl isin/Y as well as a valve z s.t. the resulting distribution of (x,y,z) is close to the distribution (D,f(D)). In this paper we initiate the study of sampling complexity, in both the classical and quantum model. We give several variants of the definition. We completely characterize some of these tasks, and give upper and lower bounds on others. In particular this allows us to establish an exponential gap between quantum and classical sampling complexity, for the set disjointness function. This is the first exponential gap for any task where the classical probabilistic algorithm is allowed to err.

We attempt to reconcilethe two distinct views of approximation classes: syntactic and computational. Syntactic classes such as MAX SNP permit structural results and have natural complete problems, while computational classes such as APX allow us to work with classes of problems whose approximability is well understood. Our results provide a syntactic characterization of computational classes and give a computational framework for syntactic classes. We compare the syntactically defined class MAX SNP with the computationally defined class APX and show that every problem in APX can be "placed" (i.e., has approximation-preserving reduction to a problem) in MAX SNP. Our methods introduce a simple, yet general, technique for creating approximation-preserving reductions which shows that any "well"-approximable problem can be reduced in an approximation-preserving manner to a problem which is hard to approximate to corresponding factors. The reduction then follows easily from the recent nonapproximability results for MAX SNP-hard problems. We demonstrate the generality of this technique by applying it to other classes such as MAX SNP-RMAX(2) and MIN F$^{+}\Pi_2(1)$ which have the clique problem and the set cover problem, respectively, as complete problems. The syntactic nature of MAX SNP was used by Papadimitriou and Yannakakis [J. Comput. System Sci., 43 (1991), pp. 425--440] to provide approximation algorithms for every problem in the class. We provide an alternate approach to demonstrating this result using the syntactic nature of MAX SNP. We develop a general paradigm, nonoblivious local search, useful for developing simple yet efficient approximation algorithms. We show that such algorithms can find good approximations for all MAX SNP problems, yielding approximation ratios comparable to the best known for a variety of specific MAX SNP-hard problems. Nonoblivious local search provably outperforms standard local search in both the degree of approximation achieved and the efficiency ofthe resulting algorithms.

Recently a great deal of attention has been focused on quantum computation following a sequence of results [Bernstein and Vazirani, in Proc. 25th Annual ACM Symposium Theory Comput., 1993, pp. 11--20, SIAM J. Comput., 26 (1997), pp. 1277--1339], [Simon, in Proc. 35th Annual IEEE Symposium Foundations Comput. Sci., 1994, pp. 116--123, SIAM J. Comput., 26 (1997), pp. 1340--1349], [Shor, in Proc. 35th Annual IEEE Symposium Foundations Comput. Sci., 1994, pp. 124--134] suggesting that quantum computers are more powerful than classical probabilistic computers. Following Shor's result that factoring and the extraction of discrete logarithms are both solvable in quantum polynomial time, it is natural to ask whether all of $\NP$ can be efficiently solved in quantum polynomial time. In this paper, we address this question by proving that relative to an oracle chosen uniformly at random with probability 1 the class $\NP$ cannot be solved on a quantum Turing machine (QTM) in time $o(2^{n/2})$. We also show that relative to a permutation oracle chosen uniformly at random with probability 1 the class $\NP \cap \coNP$ cannot be solved on a QTM in time $o(2^{n/3})$. The former bound is tight since recent work of Grover [in {\it Proc.\ $28$th Annual ACM Symposium Theory Comput.}, 1996] shows how to accept the class $\NP$ relative to any oracle on a quantum computer in time $O(2^{n/2})$.

In this paper we study quantum computation from a complexity theoretic viewpoint. Our first result is the existence of an efficient universal quantum Turing machine in Deutsch's model of a quantum Turing machine (QTM) [Proc. Roy. Soc. London Ser. A, 400 (1985), pp. 97--117]. This construction is substantially more complicated than the corresponding construction for classical Turing machines (TMs); in fact, even simple primitives such as looping, branching, and composition are not straightforward in the context of quantum Turing machines. We establish how these familiar primitives can be implemented and introduce some new, purely quantum mechanical primitives, such as changing the computational basis and carrying out an arbitrary unitary transformation of polynomially bounded dimension. We also consider the precision to which the transition amplitudes of a quantum Turing machine need to be specified. We prove that $O(\log T)$ bits of precision suffice to support a T step computation. This justifies the claim that the quantum Turing machine model should be regarded as a discrete model of computation and not an analog one. We give the first formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church--Turing thesis. We show the existence of a problem, relative to an oracle, that can be solved in polynomial time on a quantum Turing machine, but requires superpolynomial time on a bounded-error probabilistic Turing machine, and thus not in the class $\BPP$. The class $\BQP$ of languages that are efficiently decidable (with small error-probability) on a quantum Turing machine satisfies $\BPP \subseteq \BQP \subseteq \Ptime^{\SP}$. Therefore, there is no possibility of giving a mathematical proof that quantum Turing machines are more powerful than classical probabilistic Turing machines (in the unrelativized setting) unless there is a major breakthrough in complexity theory.

The rapid evolution of computers in the half century since their invention has resulted in dramatically smaller and faster computers. However, from a computational point of view, all these computers look alike; for example, they are built out of simple logic gates. A fundamental thesis of computer science---the modern form of the Church--Turing thesis---asserts that this is inevitable in a deep sense. Any computer can be simulated with at most a polynomial factor slowdown by a probabilistic Turing machine. Quantum computation poses the first credible challenge to this thesis. It goes back to a suggestion by Feynman [4], who pointed out that there appears to be no efficient way of simulating a quantum mechanical system on a computer, and suggested that, perhaps, a computer based on quantum physical principles might be able to carry out the simulation efficiently. Two formal models for quantum computers---the quantum Turing machine [2] and quantum computational networks [3]---were defined by Deutsch. The first three papers in this issue describe efficient quantum algorithms for computational tasks that we do not know how to solve classically. In "Quantum Complexity Theory," Bernstein and Vazirani give the first formal evidence that quantum computers violate the modern form of the Church--Turing thesis. They show that a certain problem---the recursive Fourier sampling problem---can be solved in polynomial time on a quantum Turing machine, but relative to an oracle, requires superpolynomial time on a classical probabilistic Turing machine. Simon, in the paper "On the Power of Quantum Computation" introduces a fundamental projection technique and uses it to design an efficient quantum algorithm to determine whether a certain type of function is 2-1 or 1-1. He further shows that, relative to an oracle, this problem requires exponential time on a classical probabilistic Turing machine. In the paper "Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer," Shor gives remarkable polynomial time quantum algorithms for two of the most famous problems in computer science: factoring and discrete log. Since the computational hardness of these problems is the basis of several famous cryptosystems, Shor's paper very dramatically underlines the power of quantum computers. To understand the computational power of quantum computers, it is helpful to consider a quantum mechanical system of n particles, each of which can be in one of two states, labeled $|0\rangle$ and $|1\rangle$. If this were a classical system, then its instantaneous state could be described by n bits. However, in quantum physics, the system is allowed to be in a linear superposition of configurations, and indeed the instantaneous state of the system is described by a unit vector in the 2n dimensional vector space, whose basis vectors correspond to all the2n classical configurations. Therefore, to describe the instantaneous state of the system, we must specify 2n complex numbers. Nature must update 2n complex numbers at each instant to evolve the system in time. This is an extraordinary amount of effort, since even for n = 200, 2n is larger than estimates of the number of elementary particles in the visible universe. Nonetheless, there are limits to the power of quantum computers. In "Strengths and Weaknesses of Quantum Computing," Bennett, Bernstein, Brassard, and Vazirani show that, relative to a random oracle, with probability 1, the class NP cannot be solved on a quantum Turing machine in time o(2n/2). This bound is tight, since recent work of Grover [5] has shown how to accept any language in NP in time O(2n/2) on a quantum Turing machine. Quantum computers are necessarily time reversible. Indeed, Bennett's work [1] on reversible computation inspired early work on quantum computation that preceded Feynman's paper [4]. The reversibility requirement makes it quite complex to implement even basic computational primitives such as looping or composition. In "Quantum Complexity Theory," Bernstein and Vazirani show how to implement quantum programming primitives and give a construction for an efficient universal quantum Turing machine. The structure of the universal quantum Turing machine is quite simple: it consists of a deterministic Turing machine with a single "quantum coin flip." In "Quantum Computability," Adleman, DeMarrais, and Huang greatly simplify this further by showing that a very simple type of coin flip is sufficient---a rotation by an angle $\theta$ such that ${\rm sin} \theta = 3/5$. Making quantum computers robust against noise and decoherence is an important and challenging problem. In "Stabilization of Quantum Computations by Symmetrization," Barenco, Berthiaume, Deutsch, Ekert, Jozsa, and Macchiavello show how to use the quantum watchdog effect to stabilize a quantum computation against noise. Their method is based on running several copies of the quantum computer in parallel and projecting its state into the symmetric subspace at frequent intervals. They show that the quantum watchdog effect results in the suppression of errors that lie outside the symmetric subspace. Quantum computation touches upon the foundations of both computer science and quantum physics. It is not unlikely that the issues raised by quantum computation will stimulate further research into the foundations of quantum physics. I wish to express my gratitude to several people who made this special section possible. Oded Goldreich acted as editor for two of the papers in the issue and dealt with them with his characteristic efficiency and judgment. The editorial staff at SIAM, most notably Lisa Dougherty, Beth Gallagher, Deidre Wunderlich, and Sam Young, were extremely helpful, patient, and resourceful. Finally, I would like to thank a number of referees whose careful and timely reviews were critical to putting together this issue. Umesh Vazirani

We can view certain randomized optimization algorithms as rules for randomly moving a particle around in a state space; each state might correspond to a distinct solution to the optimization problem, or more generally, the state space might express some other structure underlying the optimization algorithm. In this setting, a general paradigm for designing heuristics is to run several simulations of the algorithm simultaneously, and every so often classify the particles as "doing well" or "doing badly", and move each particle that is "doing badly" to the position of one that is "doing well". In this paper, we give a rigorous analysis of such a "go with the winners" scheme in the concrete setting of searching for a deep leaf in a tree. There are two relevant parameters of the tree: its depth d, and another parameter /spl kappa/ which is a measure of the imbalance of the tree. We prove that the running time of the "go with the winners" scheme (to achieve 99% probability of success) is bounded by a polynomial in d and /spl kappa/. By contrast, the simple restart scheme: run several independent simulations and pick the deepest leaf encountered takes time exponential in /spl kappa/ and d in the worst-case. We also show that any algorithm that guarantees a constant probability of success must have worst case running time at least /spl kappa/d.>

We attempt to reconcile the two distinct views of approximation classes: syntactic and computational. Syntactic classes such as MAX SNP permit structural results and have natural complete problems, while computational classes such as APX allow us to work with classes of problems whose approximability is well-understood. Our results provide a syntactic characterization of computational classes, and give a computational framework for syntactic classes.>

Quadratic Dynamical Systems (QDS), whose definition extends that of Markov chains, are used to model phenomena in a variety of fields like statistical physics and natural evolution. Such systems also play a role in genetic algorithms, a widelyused class of heuristics that are notoriously hard to analyze. Recently Rabinovich et al. took an important step in the study of QDS’s by showing, under some technical assumptions, that such systems converge to a stationary distribution (similar theorems for Markov Chains are well-known). We show, however, that the following sampling problem for QDS’s is PSPACE-hard: Given an initial distribution, produce a random sample from the t’th generation. The hardness result continues to hold for very restricted classes of QDS’s with very simple initial distributions, thus suggesting that QDS’s are intrinsically more complicated than Markov chains. ∗Supported by an IBM Graduate Fellowship and partly under NSF grant CCR-9310214. Email: [email protected]. †Work done while at ICSI, Berkeley, and supported in part by a Rothschild postdoctoral fellowship. Email: [email protected]. ‡Supported by NSF grant CCR-9310214. Email: [email protected].

In this paper, we study the leader election problem in the full information model. We show two results in this context. First, we exhibit a constructive O(log N) round protocol that is resilient against linear size coalitions. That is, our protocol is resilient against any coalition of size less then N for some constant (but small) value of. Second, we provide an easy, non-constructive probabilistic argument that shows the existence of O(log N) round protocol in which can be made as large as 1, for any positive. Our 2 protocols are extremely simple.

We study the problem of inferring an accurate model for a stochastic process from its output.We identify two desirable properties -resoluteness and reliability -of any identification algorithm.We prove that for any countable class of stochastic processes, there is an identification algorithm that has these properties.This result also formulates an optimization problem whose solution is sufficent to solve the identification problem.In this sense, our result provides an analogue to the Occam principle in a probabilistic setting.

Abstract. In this paper we study quantum computation from a complexity theoretic viewpoint. Our rst result is the existence of an ecient universal quantum Turing machine in Deutsch’s model of a quantum Turing machine (QTM) [Proc. Roy. Soc. London Ser. A, 400 (1985), pp. 97{117]. This construction is substantially more complicated than the corresponding construction for classical Turing machines (TMs); in fact, even simple primitives such as looping, branching, and composition are not straightforward in the context of quantum Turing machines. We establish how these familiar primitives can be implemented and introduce some new, purely quantum mechanical primitives, such as changing the computational basis and carrying out an arbitrary unitary transformation of polynomially bounded dimension. We also consider the precision to which the transition amplitudes of a quantum Turing machine need to be specied. We prove that O(log T) bits of precision suce to support a T step computation. This justies the claim that the quantum Turing machine model should be regarded as a discrete model of computation and not an analog one. We give the rst formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church{Turing thesis. We show the existence of a problem, relative to an oracle, that can be solved in polynomial time on a quantum Turing machine, but requires superpolynomial time on a bounded-error probabilistic Turing machine, and thus not in the class BPP. The class BQP of languages that are eciently decidable (with small error-probability) on a quantum Turing machine satises BPP ⊆ BQP ⊆ P]P. Therefore, there is no possibility of giving a mathematical proof that quantum Turing machines are more powerful than classical probabilistic Turing machines (in the unrelativized setting) unless there is a major breakthrough in complexity theory.

An approximation algorithm for the permanent of an n*n 0,1-matrix is presented. The algorithm is shown to have worst-case time complexity exp (0(n/sup 1/2/ log/sup 2/ n)). Asymptotically, this represents a considerable improvement over the best existing algorithm, which has worst-case time complexity of the form e/sup theta (n)/.>

A model of learning that expands on the Valiant model is introduced. The point of departure from the Valiant model is that the learner is placed in a Markovian environment. The environment of the learner is a (exponentially large) graph, and the examples reside on the vertices of the graph, one example on each vertex. The learner obtains the examples while performing a random walk on the graph. At each step, the learning algorithm guesses the classification of the example on the current vertex using its current hypothesis. If its guess is incorrect, the learning algorithm updates its current working hypothesis. The performance of the learning algorithm in a given environment is judged by the expected number of mistakes made as a function of the number of steps in the random walk. The predictive value of Occam algorithms under this weaker probabilistic model of the learner's environment is studied.>

Article Free Access Share on An optimal algorithm for on-line bipartite matching Authors: R. M. Karp University of California at Berkeley & International Computer Science Institute University of California at Berkeley & International Computer Science InstituteView Profile , U. V. Vazirani Cornell University Cornell UniversityView Profile , V. V. Vazirani View Profile Authors Info & Claims STOC '90: Proceedings of the twenty-second annual ACM symposium on Theory of ComputingApril 1990Pages 352–358https://doi.org/10.1145/100216.100262Published:01 April 1990Publication History 439citation3,555DownloadsMetricsTotal Citations439Total Downloads3,555Last 12 Months519Last 6 weeks66 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

The problem of computing the chromatic number of a graph is considered. No known approximation algorithm can guarantee a better than O(n/sup 0.4/) coloring on a three-chromatic graph with n vertices. Evidence is provided that it is inherently impossible to achieve a better than n/sup epsilon / ratio in polynomial time by showing that 'breaking the n/sup epsilon / barrier' will automatically lead to vastly better polynomial-time approximation algorithms that achieve ratios closer to log n.>

An efficient parallel algorithm $({\text{RNC}}^2 )$ for the two-processor scheduling problem is presented. An interesting feature of this algorithm is that it finds a highest-level-first schedule; such a schedule defines a lexicographically first solution to this problem in a natural way. A key ingredient of the algorithm is a generalization of a theorem of Tutte which establishes a one-to-one correspondence between the bases of the Tutte matrix of a graph and the sets of matched nodes in maximum matchings in the graph.

Randomized algorithms for approximating the number of perfect matchings in a graph are considered. An algorithm that is a natural simplification of one suggested and analyzed previously is introduced and analyzed. One of the key ideas is to view the analysis from a geometric perspective: it is proved that for any graph G the k-slice of the well-known Edmonds matching polytope has magnification 1. For a bipartite graph G=(U, V, E), mod U mod = mod V mod =n, with d edge-disjoint perfect matchings, it is proved that the ratio of the number of almost perfect matchings to the number of perfect matchings is at most n/sup 3n/d/. For any constant alpha >0 this yields a a fully polynomial randomized algorithm for approximating the number of perfect matchings in bipartite graphs with d>or= alpha n. Moreover, for some constant c>0 it is the fastest known approximation algorithm for bipartite graphs with d>or= clog n.>

A new algorithm for finding a maximum matching in a general graph is presented; its special feature being that the only computationally non-trivial step required in its execution is the inversion of a single integer matrix.Since this step can be parallelized, we get a simple parallel (RNC2) algorithm.At the heart of our algorithm lies a probabilistic lemma, the isolating lemma.We show applications of this lemma to parallel computation and randomized reductions.

Article Free Access Share on Efficiency considerations in using semi-random sources Author: U. Vazirani Harvard University, Cambridge, MA Harvard University, Cambridge, MAView Profile Authors Info & Claims STOC '87: Proceedings of the nineteenth annual ACM symposium on Theory of computingJanuary 1987 Pages 160–168https://doi.org/10.1145/28395.28413Online:01 January 1987Publication History 44citation320DownloadsMetricsTotal Citations44Total Downloads320Last 12 Months8Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Random Polynomial Time (Rp) is currently considered to be the class of tractable computational problems. Here one assumes a source of truly random bits. However, the known sources of randomness are imperfect. They can be modeled as an adversary source, called slightly-random source. Slightlyrandom Polynomial Time (SRp) is the class of problems solvable in polynomial time using such a source. SRp is thus a more realistic definition of a tractable computational problem. In this paper we give an affirmative answer to the question "is Rp = SRp?" Our proof method is constructive: given an Rp algorithm for a problem, we show how to obtain an SRp algorithm for it. Studying the relationship between randomized and deterministic computation is currently an important issue. A central question here is "is Rp = P?" Our result may be a step towards answering this question.

Article Free Access Share on Towards a strong communication complexity theory or generating quasi-random sequences from two communicating slightly-random sources Author: U V Vazirani University of California, Berkeley, CA University of California, Berkeley, CAView Profile Authors Info & Claims STOC '85: Proceedings of the seventeenth annual ACM symposium on Theory of computingDecember 1985 Pages 366–378https://doi.org/10.1145/22145.22186Online:01 December 1985Publication History 32citation372DownloadsMetricsTotal Citations32Total Downloads372Last 12 Months8Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

The two-processor scheduling problem is perhaps the most basic problem in scheduling theory, and several efficient algorithms have been discovered for it. However, these algorithms are inherently sequential in nature. We give a fast parallel (R-NC) algorithm for this problem. Interestingly enough, our algorithm for this purely combinatoric-looking problem draws on some powerful algebraic methods.

Several applications require truly random bit sequences, whereas physical sources of randomness are at best imperfect. We consider a general model for these slightly-random sources (e,g. zener diodes), and show how to convert their output into 'random looking ' sequences, which we call quasi -random. We show that quasi-random sequences are indistinguishable from truly random ones in a strong sense. This enables us to prove that quasi-random sequences can be used in place of truly random ones for applications such as seeds for pseudo-random number generators, randomizing algorithms, and stochastic simulation experiments.

Cryptographically secure pseudo-random number generators known so far suffer from the handicap of being inefficient; the most efficient ones can generate only one bit on each modular multiplication (n/sup 2/ steps). Blum, Blum and Shub ask the open problem of outputting even two bits securely. We state a simple condition, the XOR-Condition, and show that any generator satisfying this condition can output logn bits on each multiplication. We also show that the logn least significant bits of RSA, Rabin's Scheme, and the x/sup 2/ mod N generator satisfy boolean predicates of these bits are secure. Furthermore, we strengthen the security of the x/sup 2/ mod N generator, which being a Trapdoor Generator, has several applications, by proving it as hard as Factoring.

We examine the problem of routing wires on a VLSI chip, where the pins to be connected are arranged in a regular rectangular array. We obtain tight bounds for the worst-case "channel-width" needed to route an n × n array, and develop provably good heuristics for the general case. An interesting "rounding algorithm" for obtaining integral approximations to solutions of linear equations is used to show the near-optimality of single-turn routings in the worst-case.

We define the class of trapdoor pseudo-random number generators, and introduce a new technique for using these in cryptography. As an application for this technique, we present a provably secure protocol for One-Bit Disclosures i.e. for giving a one-bit message in exchange for receipt.

We prove a natural encoding scheme intractable (by showing it UR-complete, a technique which may be used when a problem does not yield to a proof of NP-completeness). This is the first non number-theoretic problem that is UR-complete but not known to be NP-complete. We also redefine UR-completeness (henceforth refered to as PR-completeness) in probabilistic terms thus making the notion conceptually simpler. Our result suggests that PR-completeness may be a more widely applicable technique than was previously believed.