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P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · n = 10

Choose n nonzero vectors in C^d — n points of complex projective space — minimizing the largest normalized Hermitian overlap μ = max |⟨z_i, z_j⟩| / (|z_i||z_j|).

Instanced = 6, n = 10
ObjectiveMinimize the largest coherence μ
Best known, unproven0.272203145the Game of Sloanes leader board (the table of putatively optimal packings maintained by Jasper, King and Mixon)

Formal definition

  • ContainerComplex d-space C^d; an answer is n points of complex projective space CP^{d-1}
  • SubmissionExactly n nonzero complex vectors, each d pairs [re, im]
  • ObjectiveMinimize μ² = max |⟨z_i,z_j⟩|²/(|z_i|²|z_j|²); moduli and norms squared are rational, compared exactly by cross-multiplication
  • ScoringThe record is ceil(μ²·10¹⁸); the page shows μ, rounded up at the ninth decimal
Challenge this record ↗
VERIFIED CONSTRUCTIONpairwise overlap heat of 10 directions; brighter is closer

Getting a feel for it

Where it is used

Complex projective codebooks are quantum measurements (the SIC-POVM family), communication codebooks, and noise-robust data representations. The further apart the codewords, the better they survive noise and erasures.

Where the frontier is

The Game of Sloanes is a public leader board of putatively optimal packings that explicitly invites improvement. Every sub-problem here is chosen from its still-open rows — the ones with a real gap between the best known value and the lower bound.

Source
Current leader

0.994780215

the largest coherence μ

Record holderFounding benchmark
Solution methodHuman
Challenge this record ↗
ANSWER FORMAT

How to write your answer

Each complex vector is d pairs [re, im], components written as decimal strings in [-1, 1] with at most nine decimal places. Global phase and nonzero complex scaling represent the same projective point.

Submit vectors: exactly n rows of d pairs [re, im], both decimal strings in [-1, 1]. Each row is a nonzero complex vector; global phase and nonzero complex scaling do not change the answer.

The current leader's answer

{
  "vectors": [
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.100000000",
        "0"
      ],
      [
        "0.010000000",
        "0"
      ],
      [
        "0.001000000",
        "0"
      ],
      [
        "0.000100000",
        "0"
      ],
      [
        "0.000010000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.200000000",
        "0"
      ],
      [
        "0.040000000",
        "0"
      ],
      [
        "0.008000000",
        "0"
      ],
      [
        "0.001600000",
        "0"
      ],
      [
        "0.000320000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.300000000",
        "0"
      ],
      [
        "0.090000000",
        "0"
      ],
      [
        "0.027000000",
        "0"
      ],
      [
        "0.008100000",
        "0"
      ],
      [
        "0.002430000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.400000000",
        "0"
      ],
      [
        "0.160000000",
        "0"
      ],
      [
        "0.064000000",
        "0"
      ],
      [
        "0.025600000",
        "0"
      ],
      [
        "0.010240000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.500000000",
        "0"
      ],
      [
        "0.250000000",
        "0"
      ],
      [
        "0.125000000",
        "0"
      ],
      [
        "0.062500000",
        "0"
      ],
      [
        "0.031250000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.600000000",
        "0"
      ],
      [
        "0.360000000",
        "0"
      ],
      [
        "0.216000000",
        "0"
      ],
      [
        "0.129600000",
        "0"
      ],
      [
        "0.077760000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.700000000",
        "0"
      ],
      [
        "0.490000000",
        "0"
      ],
      [
        "0.343000000",
        "0"
      ],
      [
        "0.240100000",
        "0"
      ],
      [
        "0.168070000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.800000000",
        "0"
      ],
      [
        "0.640000000",
        "0"
      ],
      [
        "0.512000000",
        "0"
      ],
      [
        "0.409600000",
        "0"
      ],
      [
        "0.327680000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.900000000",
        "0"
      ],
      [
        "0.810000000",
        "0"
      ],
      [
        "0.729000000",
        "0"
      ],
      [
        "0.656100000",
        "0"
      ],
      [
        "0.590490000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ]
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 10,
  "d": 6
}

The current leader's answer

{
  "vectors": [
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.100000000",
        "0"
      ],
      [
        "0.010000000",
        "0"
      ],
      [
        "0.001000000",
        "0"
      ],
      [
        "0.000100000",
        "0"
      ],
      [
        "0.000010000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.200000000",
        "0"
      ],
      [
        "0.040000000",
        "0"
      ],
      [
        "0.008000000",
        "0"
      ],
      [
        "0.001600000",
        "0"
      ],
      [
        "0.000320000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.300000000",
        "0"
      ],
      [
        "0.090000000",
        "0"
      ],
      [
        "0.027000000",
        "0"
      ],
      [
        "0.008100000",
        "0"
      ],
      [
        "0.002430000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.400000000",
        "0"
      ],
      [
        "0.160000000",
        "0"
      ],
      [
        "0.064000000",
        "0"
      ],
      [
        "0.025600000",
        "0"
      ],
      [
        "0.010240000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.500000000",
        "0"
      ],
      [
        "0.250000000",
        "0"
      ],
      [
        "0.125000000",
        "0"
      ],
      [
        "0.062500000",
        "0"
      ],
      [
        "0.031250000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.600000000",
        "0"
      ],
      [
        "0.360000000",
        "0"
      ],
      [
        "0.216000000",
        "0"
      ],
      [
        "0.129600000",
        "0"
      ],
      [
        "0.077760000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.700000000",
        "0"
      ],
      [
        "0.490000000",
        "0"
      ],
      [
        "0.343000000",
        "0"
      ],
      [
        "0.240100000",
        "0"
      ],
      [
        "0.168070000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.800000000",
        "0"
      ],
      [
        "0.640000000",
        "0"
      ],
      [
        "0.512000000",
        "0"
      ],
      [
        "0.409600000",
        "0"
      ],
      [
        "0.327680000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.900000000",
        "0"
      ],
      [
        "0.810000000",
        "0"
      ],
      [
        "0.729000000",
        "0"
      ],
      [
        "0.656100000",
        "0"
      ],
      [
        "0.590490000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ]
    ]
  ]
}

Submit vectors: exactly n rows of d pairs [re, im], both decimal strings in [-1, 1]. Each row is a nonzero complex vector; global phase and nonzero complex scaling do not change the answer. · Verifier v1.0.0

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