Uncertainty relations are a distinctive characteristic of quantum theory that impose intrinsic limitations on the precision with which physical properties can be simultaneously determined. The modern work on uncertainty relations employs entropic measures to quantify the lack of knowledge associated with measuring non-commuting observables. However, there is no fundamental reason for using entropies as quantifiers; any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation. Starting from a very reasonable assumption of invariance under mere relabelling of the measurement outcomes, we show that Schur-concave functions are the most general uncertainty quantifiers. We then discover a fine-grained uncertainty relation that is given in terms of the majorization order between two probability vectors, significantly extending a majorization-based uncertainty relation first introduced in [M. H. Partovi, Phys. Rev. A 84, 052117 (2011)]. Such a vector-type uncertainty relation generates an infinite family of distinct scalar uncertainty relations via the application of arbitrary uncertainty quantifiers. Our relation is therefore universal and captures the essence of uncertainty in quantum theory.Uncertainty relations lie at the core of quantum mechanics and are a direct manifestation of the noncommutative structure of the theory. In contrast to classical physics, where in principle any observable can be measured with arbitrary precision, quantum mechanics introduces severe restrictions on the allowed measurement results of two or more non-commuting observables. Uncertainty relations are not a manifestation of the experimentalists' (in)ability of performing precise measurements, but are inherently determined by the incompatibility of the measured observables.The first formulation of the uncertainty principle was provided by Heisenberg [1], who noted that more knowledge about the position of a single quantum particle implies less certainty about its momentum and vice-versa. He expressed the principle in terms of standard deviations of the momentum and position operators ∆X · ∆P 2 .Robertson [2] generalized Heisenberg's uncertainty principle to any two arbitrary observables A and B asA major drawback of Robertson's uncertainty principle is that it depends on the state |ψ of the system. In particular, when |ψ belongs to the null-space of the * Electronic address: friedlan@uic. Here H(A) is the Shannon entropy [5] of the probability distribution induced by measuring the state |ψ of the system in the eigenbasis {|a j } of the oservable A (and similarly for B). The bound on the right hand side c(A, B) := max m,n | a m |b n | represents the maximum overlap between the bases elements, and is independent of the state |ψ . Recently the study of uncertainty relations intensified [6,7] (see also [1,9] for recent surveys), and as a result various important applications have been discovered, ranging from security proofs for quantum cryptography [10][11][12], information locking ...

Currently available quantum computing hardware platforms have limited 2-qubit connectivity among their addressable qubits. In order to run a generic quantum algorithm on such a platform, one has to transform the initial logical quantum circuit describing the algorithm into an equivalent one that obeys the connectivity restrictions.In this work we construct a circuit synthesis scheme that takes as input the qubit connectivity graph and a quantum circuit over the gate set generated by {CNOT, RZ } and outputs a circuit that respects the connectivity of the device. As a concrete application, we apply our techniques to Google's Bristlecone 72-qubit quantum chip connectivity, IBM's Tokyo 20-qubit quantum chip connectivity, and Rigetti's Acorn 19-qubit quantum chip connectivity. In addition, we also compare the performance of our scheme as a function of sparseness of randomly generated quantum circuits.

A Holevo measure is used to discuss how much information about a given POVM on system a is present in another system b, and how this influences the presence or absence of information about a different POVM on a in a third system c. The main goal is to extend information theorems for mutually unbiased bases or general bases to arbitrary POVMs, and especially to generalize "all-or-nothing" theorems about information located in tripartite systems to the case of partial information, in the form of quantitative inequalities. Some of the inequalities can be viewed as entropic uncertainty relations that apply in the presence of quantum side information, as in recent work by Berta et al. [Nature Physics 6, 659 (2010)]. All of the results also apply to quantum channels: e.g., if E accurately transmits certain POVMs, the complementary channel F will necessarily be noisy for certain other POVMs. While the inequalities are valid for mixed states of tripartite systems, restricting to pure states leads to the basis-invariance of the difference between the information about a contained in b and c.

We study the robustness of the bucket brigade quantum random access memory model introduced by Giovannetti et al (2008 Phys. Rev. Lett.100 160501). Due to a result of Regev and Schiff (ICALP '08 733), we show that for a class of error models the error rate per gate in the bucket brigade quantum memory has to be of order o 2 n 2 ( ) -(where N 2 n = is the size of the memory) whenever the memory is used as an oracle for the quantum searching problem. We conjecture that this is the case for any realistic error model that will be encountered in practice, and that for algorithms with superpolynomially many oracle queries the error rate must be super-polynomially small, which further motivates the need for quantum error correction. By contrast, for algorithms such as matrix inversion Harrow et al (2009 Phys. Rev. Lett.103 150502) or quantum machine learning Rebentrost et al (2014 Phys. Rev. Lett.113 130503) that only require a polynomial number of queries, the error rate only needs to be polynomially small and quantum error correction may not be required. We introduce a circuit model for the quantum bucket brigade architecture and argue that quantum error correction for the circuit causes the quantum bucket brigade architecture to lose its primary advantage of a small number of 'active' gates, since all components have to be actively error corrected.