Evenly distributed unitaries: On the structure of unitary designs

By David Gross, Koenraad M. R. Audenaert, Jens Eisert. 2006

395 citations recorded by the source corpus.

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.

Subfields: Electrical and Electronic Engineering, Management Science and Operations Research, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/7a09ad921b4880aae83c325c5f9d8bb36dd17fd8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Electrical and Electronic Engineering

In Management Science and Operations Research

In Materials Chemistry

Citations in the corpus

Superconducting quantum circuits at the surface code threshold for fault tolerance (2014, neutral): “A set of unitaries {Uk}k=1 is a 2-design if and only if [15]”
Geometry of Quantum States: Index (2007, background): “The idea was further developed in papers to which we refer for proofs, applications, and details [367, 778].”
Linear growth of quantum circuit complexity (2021, neutral): “[42] A t-design is a probability distribution, over unitaries, whose first t moments equal the Haar measure’s moments [17, 23].”
Corrigendum: Negative quasi-probability as a resource for quantum computation (2012, background): “This restricted set is the Clifford group, and we now review the important parts of its structure for qudit systems [27].”
Unitary designs from statistical mechanics in random quantum circuits. (2019, background): “The frame potential for an ensemble of unitaries E was ﬁrst discussed in the context of unitary designs in [23,24].”
Local random quantum circuits are approximate polynomial-designs: numerical results (2012, methodology): “[36] D.”
Negative quasi-probability as a resource for quantum computation (2012, background): “This restricted set is the Clifford group, and we now review the important parts of its structure for qudit systems (Gross et al 2007).”
Unbounded Violation of Tripartite Bell Inequalities (2007, background): “Question 3: Does our construction say something about the additivity problem [66]? The other question is motivated by the recent interest in unitary designs [ 32 ], as a way to derandomize many quantum protocols.”
The resource theory of stabilizer quantum computation (2013, methodology): “For an overview of the phase space techniques for the stabilizer formalism see [22, 24].”
Detecting crosstalk errors in quantum information processors (2019, neutral): “noise because they twirl it [14, 20, 7].”
Random Quantum Circuits are Approximate 2-designs (2008, background): “Approximate unitary 2-designs have been considered [13, 10, 17], although the approaches are specific to 2-designs.”
Onset of random matrix behavior in scrambling systems (2018, background): “In particular these ideas have been applied to the study of approximate unitary k-designs [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40].”
Optimizing quantum process tomography with unitary 2-designs (2007, background): “In the unital case, these POVMs are equivalent to weighted unitary 2-designs [10, 11], but in the general case, they are shown not to exist.”
Entanglement detection (2008, background): “The resulting Werner states are the linear combination of relatively few basis matrices [44,246,247].”
The randomized measurement toolbox (2022, methodology): “Examples of such unitary designs 38,39 include the single-qubit Clifford group and the full unitary group U (2) encompassing all single-qubit transformations.”