Quantum Query Algorithms

Elektroniskā versija nesatur pielikumusLELDE LĀCE KVANTU VAICĀJOŠIE ALGORITMI ANOTĀCIJA Kvantu skaitļošana ir datorzinātņu apakšnozare, kurā tiek izmantotas kvantu mehānikas īpatnības, lai efektīvāk risinātu skaitļošanas uzdevumus. Šajā darbā tiek aplūkoti kvantu vaicājošie algoritmi Bula funkciju rēķināšanai. Darba sākumā tiek pierādīti kvantu algoritmu apakšējie novērtējumi dažādām funkcijām, kas apraksta grafu problēmas. Promocijas darba galvenais uzdevums ir izveidot efektīvus kvantu vaicājošos algoritmus. Ir nodefinēts kā veidot precīzus kvantu vaicājošos algoritmus ar sarežģītību n-1, 2n/3 un n/2. Darba turpinājumā tiek analizēti nedeterminētie kvantu algoritmi ar vienu jautājumu, to veidošanas iespējas un īpašības. Promocijas darbā tiek definēts jauns kvantu vaicājošo algoritmu veids - kvantu vaicājošie algoritmi ar pēcatlasi un tiek pierādīta šo algoritmu saistība ar nedeterminētajiem kvantu vaicājošajiem algoritmiem.LELDE LĀCE QUANTUM QUERY ALGORITHMS ANNOTATION Quantum computing is the subfield of computer science that aims to employ effects of quantum mechanics to efficiently perform computational tasks. The main research object of this work is quantum query model to compute Boolean functions. At first we prove higher lower bounds of quantum query algorithms for some of graph problems. Main purpose of the research is to find quantum query algorithms with complexity lower than deterministic one. The work presents a set of new exact quantum algorithms with quantum query complexity n-1, 2n/3 and n/2. We construct some nondeterministic quantum query algorithms with complexity 1 for Boolean functions with 2, 4 and 2n variables and study some properties of these functions. We propose definition of postselection quantum query algorithm and we propose one method how to make postselection quantum query algorithms

The Computational Complexity of Linear Optics

We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P^#P=BPP^NP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation. Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the "Permanent-of-Gaussians Conjecture", which says that it is #P-hard to approximate the permanent of a matrix A of independent N(0,1) Gaussian entries, with high probability over A; and the "Permanent Anti-Concentration Conjecture", which says that |Per(A)|>=sqrt(n!)/poly(n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application. This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists.Comment: 94 pages, 4 figure

NP-complete Problems and Physical Reality

Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and "anthropic computing." The section on soap bubbles even includes some "experimental" results. While I do not believe that any of the proposals will let us solve NP-complete problems efficiently, I argue that by studying them, we can learn something not only about computation but also about physics.Comment: 23 pages, minor correction

Shadow Tomography of Quantum States

We introduce the problem of *shadow tomography*: given an unknown $D$-dimensional quantum mixed state $\rho$, as well as known two-outcome measurements $E_{1},\ldots,E_{M}$, estimate the probability that $E_{i}$ accepts $\rho$, to within additive error $\varepsilon$, for each of the $M$ measurements. How many copies of $\rho$ are needed to achieve this, with high probability? Surprisingly, we give a procedure that solves the problem by measuring only $\widetilde{O}\left( \varepsilon^{-4}\cdot\log^{4} M\cdot\log D\right)$ copies. This means, for example, that we can learn the behavior of an arbitrary $n$-qubit state, on all accepting/rejecting circuits of some fixed polynomial size, by measuring only $n^{O\left( 1\right)}$ copies of the state. This resolves an open problem of the author, which arose from his work on private-key quantum money schemes, but which also has applications to quantum copy-protected software, quantum advice, and quantum one-way communication. Recently, building on this work, Brand\~ao et al. have given a different approach to shadow tomography using semidefinite programming, which achieves a savings in computation time.Comment: 29 pages, extended abstract appeared in Proceedings of STOC'2018, revised to give slightly better upper bound (1/eps^4 rather than 1/eps^5) and lower bounds with explicit dependence on the dimension

Oracles Are Subtle But Not Malicious

Theoretical computer scientists have been debating the role of oracles since the 1970's. This paper illustrates both that oracles can give us nontrivial insights about the barrier problems in circuit complexity, and that they need not prevent us from trying to solve those problems. First, we give an oracle relative to which PP has linear-sized circuits, by proving a new lower bound for perceptrons and low- degree threshold polynomials. This oracle settles a longstanding open question, and generalizes earlier results due to Beigel and to Buhrman, Fortnow, and Thierauf. More importantly, it implies the first nonrelativizing separation of "traditional" complexity classes, as opposed to interactive proof classes such as MIP and MA-EXP. For Vinodchandran showed, by a nonrelativizing argument, that PP does not have circuits of size n^k for any fixed k. We present an alternative proof of this fact, which shows that PP does not even have quantum circuits of size n^k with quantum advice. To our knowledge, this is the first nontrivial lower bound on quantum circuit size. Second, we study a beautiful algorithm of Bshouty et al. for learning Boolean circuits in ZPP^NP. We show that the NP queries in this algorithm cannot be parallelized by any relativizing technique, by giving an oracle relative to which ZPP^||NP and even BPP^||NP have linear-size circuits. On the other hand, we also show that the NP queries could be parallelized if P=NP. Thus, classes such as ZPP^||NP inhabit a "twilight zone," where we need to distinguish between relativizing and black-box techniques. Our results on this subject have implications for computational learning theory as well as for the circuit minimization problem.Comment: 20 pages, 1 figur