skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Evenly distributed unitaries: On the structure of unitary designs

Abstract

We clarify the mathematical structure underlying unitary t-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group. We present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design. Lower bounds for the number of elements of 2-designs are derived. We show how to turn mutually unbiased bases into approximate 2-designs whose cardinality is optimal in leading order. Designs of higher order are discussed and an example of a unitary 5-design is presented. We comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups. Further, we sketch connections to problems in linear optics and questions regarding typical entanglement.

Authors:
; ;  [1]
+ Show Author Affiliations
  1. Institute for Mathematical Sciences, Imperial College London, Princes Gate, London SW7 2PE (United Kingdom) and QOLS, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BW (United Kingdom)
Publication Date:
OSTI Identifier:
20929700
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 5; Other Information: DOI: 10.1063/1.2716992; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; GROUP THEORY; INFORMATION THEORY; MATRICES; POLYNOMIALS; QUANTUM ENTANGLEMENT; QUANTUM MECHANICS

Citation Formats

Gross, D., Audenaert, K., and Eisert, J. Evenly distributed unitaries: On the structure of unitary designs. United States: N. p., 2007. Web. doi:10.1063/1.2716992.
Copy to clipboard
Gross, D., Audenaert, K., & Eisert, J. Evenly distributed unitaries: On the structure of unitary designs. United States. doi:10.1063/1.2716992.
Copy to clipboard
Gross, D., Audenaert, K., and Eisert, J. Tue . "Evenly distributed unitaries: On the structure of unitary designs". United States. doi:10.1063/1.2716992.
Copy to clipboard
@article{osti_20929700,
title = {Evenly distributed unitaries: On the structure of unitary designs},
author = {Gross, D. and Audenaert, K. and Eisert, J.},
abstractNote = {We clarify the mathematical structure underlying unitary t-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group. We present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design. Lower bounds for the number of elements of 2-designs are derived. We show how to turn mutually unbiased bases into approximate 2-designs whose cardinality is optimal in leading order. Designs of higher order are discussed and an example of a unitary 5-design is presented. We comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups. Further, we sketch connections to problems in linear optics and questions regarding typical entanglement.},
doi = {10.1063/1.2716992},
journal = {Journal of Mathematical Physics},
number = 5,
volume = 48,
place = {United States},
year = {Tue May 15 00:00:00 EDT 2007},
month = {Tue May 15 00:00:00 EDT 2007}
}
Copy to clipboard
Journal Article:
DOI:  10.1063/1.2716992
Other availability
Find in Google Scholar Find in Google Scholar
Search WorldCat Search WorldCat to find libraries that may hold this journal
Save / Share:
Export Metadata
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.

  • ; ; ; ... - Physical Review. A
    We develop the concept of a unitary t-design as a means of expressing operationally useful subsets of the stochastic properties of the uniform (Haar) measure on the unitary group U(2{sup n}) on n qubits. In particular, sets of unitaries forming 2-designs have wide applicability to quantum information protocols. We devise an O(n)-size in-place circuit construction for an approximate unitary 2-design. We then show that this can be used to construct an efficient protocol for experimentally characterizing the fidelity of a quantum process on n qubits with quantum circuits of size O(n) without requiring any ancilla qubits, thereby improving upon previousmore » approaches.« less

  • ; ; ; ... - Biofuels, Bioproducts and Biorefining

  • ; ; ; ... - Biofuels, Bioproducts & Biorefining
    The 2011 US Billion-Ton Update estimates that by 2030 there will be enough agricultural and forest resources to sustainably provide at least one billion dry tons of biomass annually, enough to displace approximately 30% of the country's current petroleum consumption. A portion of these resources are inaccessible at current cost targets with conventional feedstock supply systems because of their remoteness or low yields. Reliable analyses and projections of US biofuels production depend on assumptions about the supply system and biorefinery capacity, which, in turn, depend upon economic value, feedstock logistics, and sustainability. A cross-functional team has examined combinations of advancesmore » in feedstock supply systems and biorefinery capacities with rigorous design information, improved crop yield and agronomic practices, and improved estimates of sustainable biomass availability. A previous report on biochemical refinery capacity noted that under advanced feedstock logistic supply systems that include depots and pre-processing operations there are cost advantages that support larger biorefineries up to 10 000 DMT/day facilities compared to the smaller 2000 DMT/day facilities. This report focuses on analyzing conventional versus advanced depot biomass supply systems for a thermochemical conversion and refinery sizing based on woody biomass. The results of this analysis demonstrate that the economies of scale enabled by advanced logistics offsets much of the added logistics costs from additional depot processing and transportation, resulting in a small overall increase to the minimum ethanol selling price compared to the conventional logistic supply system. While the overall costs do increase slightly for the advanced logistic supply systems, the ability to mitigate moisture and ash in the system will improve the storage and conversion processes. In addition, being able to draw on feedstocks from further distances will decrease the risk of biomass supply to the conversion facility.« less

  • ; ; ; ... - Biofuels, Bioproducts & Biorefining
    The 2011 US Billion-Ton Update estimates that by 2030 there will be enough agricultural and forest resources to sustainably provide at least one billion dry tons of biomass annually, enough to displace approximately 30% of the country's current petroleum consumption. A portion of these resources are inaccessible at current cost targets with conventional feedstock supply systems because of their remoteness or low yields. Reliable analyses and projections of US biofuels production depend on assumptions about the supply system and biorefinery capacity, which, in turn, depend upon economic value, feedstock logistics, and sustainability. A cross-functional team has examined combinations of advancesmore » in feedstock supply systems and biorefinery capacities with rigorous design information, improved crop yield and agronomic practices, and improved estimates of sustainable biomass availability. A previous report on biochemical refinery capacity noted that under advanced feedstock logistic supply systems that include depots and pre-processing operations there are cost advantages that support larger biorefineries up to 10 000 DMT/day facilities compared to the smaller 2000 DMT/day facilities. This report focuses on analyzing conventional versus advanced depot biomass supply systems for a thermochemical conversion and refinery sizing based on woody biomass. The results of this analysis demonstrate that the economies of scale enabled by advanced logistics offsets much of the added logistics costs from additional depot processing and transportation, resulting in a small overall increase to the minimum ethanol selling price compared to the conventional logistic supply system. While the overall costs do increase slightly for the advanced logistic supply systems, the ability to mitigate moisture and ash in the system will improve the storage and conversion processes. In addition, being able to draw on feedstocks from further distances will decrease the risk of biomass supply to the conversion facility.« less

  • ; ; ; ... - Phys. Rev., 161: 1015-27 (Sept. 20, 1967).