Quark confinement and the properties of hadrons

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A lower bound on the quantum query complexity of read-once functions

We establish a lower bound of $\Omega{(\sqrt{n})}$ on the bounded-error quantum query complexity of read-once Boolean functions, providing evidence for the conjecture that $\Omega(\sqrt{D(f)})$ is a lower bound for all Boolean functions. Our technique extends a result of Ambainis, based on the idea that successful computation of a function requires ``decoherence'' of initially coherently superposed inputs in the query register, having different values of the function. The number of queries is bounded by comparing the required total amount of decoherence of a judiciously selected set of input-output pairs to an upper bound on the amount achievable in a single query step. We use an extension of this result to general weights on input pairs, and general superpositions of inputs.Comment: 12 pages, LaTe

Regional VARs and the channels of monetary policy

We find that the magnitudes of the regional effects of monetary policy were considerably dampened during the Volcker-Greenspan era. Further, regional differences in the depths of monetary-policy-induced recessions are related to the concentration of the banking sector, whereas differences in the total cost of these recessions are related to industry mix.Monetary policy

Lower bounds on the non-Clifford resources for quantum computations

We establish lower-bounds on the number of resource states, also known as magic states, needed to perform various quantum computing tasks, treating stabilizer operations as free. Our bounds apply to adaptive computations using measurements and an arbitrary number of stabilizer ancillas. We consider (1) resource state conversion, (2) single-qubit unitary synthesis, and (3) computational tasks. To prove our resource conversion bounds we introduce two new monotones, the stabilizer nullity and the dyadic monotone, and make use of the already-known stabilizer extent. We consider conversions that borrow resource states, known as catalyst states, and return them at the end of the algorithm. We show that catalysis is necessary for many conversions and introduce new catalytic conversions, some of which are close to optimal. By finding a canonical form for post-selected stabilizer computations, we show that approximating a single-qubit unitary to within diamond-norm precision $\varepsilon$ requires at least $1/7\cdot\log_2(1/\varepsilon) - 4/3$ $T$-states on average. This is the first lower bound that applies to synthesis protocols using fall-back, mixing techniques, and where the number of ancillas used can depend on $\varepsilon$. Up to multiplicative factors, we optimally lower bound the number of $T$ or $CCZ$ states needed to implement the ubiquitous modular adder and multiply-controlled-$Z$ operations. When the probability of Pauli measurement outcomes is 1/2, some of our bounds become tight to within a small additive constant.Comment: 62 page

Merger Policy and Innovation: Must Enforcement Change to Account for Technological Change?

Merger policy is the most active area of U.S. antitrust policy. It is now widely believed that merger policy must move beyond its traditional focus on static efficiency to account for innovation and address dynamic efficiency. Innovation can fundamentally affect merger analysis in two ways. First, innovation can dramatically affect the relationship between the pre-merger marketplace and what is likely to happen if a proposed merger is consummated. Thus, innovation can fundamentally influence the appropriate analysis for addressing traditional, static efficiency concerns. Second, innovation can itself be an important dimension of market performance that is potentially affected by a merger. We explore how merger policy is meeting the challenges posed by innovation.

Structural breaks and regional disparities in the transmission of monetary policy

Using a regional VAR, we find large differences in the effects of monetary policy shocks across regions of the United States. We also find that the region-level effects of monetary policy differ a great deal between the pre-Volcker and Volcker-Greenspan periods in terms of their depth and length. The two sample periods also yield very different rankings of the regions in terms of the effects of monetary policy. Our regional VAR also suggests that aggregate VARs that ignore regional variations can suffer from severe aggregation bias. We use the results of our regional VAR to find evidence that recession depth related to the banking concentration and that the total cost of recession is related to the industry mix. Finally, we demonstrate that the differences between the two sample periods are due to changes in the mechanism by which monetary policy shocks are propagated.Monetary policy ; Regional economics ; Economic conditions