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DESCRIPTION:Quantum colorings of spheres\nOlivier Lalonde\nCameron\, Montanaro\, Newman\, Severini and Winter gave a construction which shows that\, for n in {2\,4\,8}\, any graph G which admits a real n-dimensional orthogonal representation is quantumly n-colorable. This result can be recast as the statement that the real sphere S^{n-1} is quantumly n-colorable for these values of n. We investigate possible extensions of their construction. We first show that their hypothesis that the orthogonal representation be real-valued is required by proving that there is no analogue of this for the complex spheres\, which all have quantum chromatic number strictly bigger than the dimension except in two dimensions. We also provide finitary witnesses of this and show for the first time that the real and complex orthogonal ranks are distinct as a byproduct. For the real case\, we show that if S^{n-1} is quantumly n-colorable\, then either n=2 or n is a multiple of 4\, and show that the converse holds whenever a Hadamard matrix of order n exists. Hence\, assuming the Hadamard conjecture\, this completely classifies the dimensions to which the CMNSW construction can be extended. Our method of proof involves showing the equivalence between the existence of such a construction and the ability to find a maximal code space for Clifford-algebraic errors given a clean ancilla\, and we believe that the representation-theoretic techniques we use for tackling the latter problem could be of independent interest. It also follows from this equivalence that S^{n-1} admits a rank-one quantum n-coloring if and only if n in {2\,4\,8}\, thereby settling a conjecture of Zeng and Zhang. \nLocation \nQNC 1201
X-ALT-DESC;FMTTYPE=text/html:Quantum colorings of spheres<br />Olivier Lalonde<br />Cameron, Montanaro, Newman, Severini and Winter gave a construction which shows that, for n in {2,4,8}, any graph G which admits a real n-dimensional orthogonal representation is quantumly n-colorable. This result can be recast as the statement that the real sphere S^{n-1} is quantumly n-colorable for these values of n. We investigate possible extensions of their construction. We first show that their hypothesis that the orthogonal representation be real-valued is required by proving that there is no analogue of this for the complex spheres, which all have quantum chromatic number strictly bigger than the dimension except in two dimensions. We also provide finitary witnesses of this and show for the first time that the real and complex orthogonal ranks are distinct as a byproduct. For the real case, we show that if S^{n-1} is quantumly n-colorable, then either n=2 or n is a multiple of 4, and show that the converse holds whenever a Hadamard matrix of order n exists. Hence, assuming the Hadamard conjecture, this completely classifies the dimensions to which the CMNSW construction can be extended. Our method of proof involves showing the equivalence between the existence of such a construction and the ability to find a maximal code space for Clifford-algebraic errors given a clean ancilla, and we believe that the representation-theoretic techniques we use for tackling the latter problem could be of independent interest. It also follows from this equivalence that S^{n-1} admits a rank-one quantum n-coloring if and only if n in {2,4,8}, thereby settling a conjecture of Zeng and Zhang. <br />Location <br />QNC 1201
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SUMMARY:IQC Quantum Catalyst seminar featuring Olivier Lalonde
DTSTART;TZID=America/New_York:20260611T163000
DTEND;TZID=America/New_York:20260611T173000
DTSTAMP:20260606T152805Z
TRANSP:OPAQUE
STATUS:CONFIRMED
SEQUENCE:0
LOCATION:QNC 1201
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