Blind Symmetry Matching in Quantum States with Application to Shot-Count Reduction

Abstract: Measuring a quantum computation in a basis adapted to a symmetry it carries reduces the repeated measurements, commonly referred to as ``shots'', needed to read a statistical answer. Detecting the symmetry a quantum state carries has many uses: certifying a claimed symmetry, identifying a conserved-charge sector, flagging symmetry-breaking as an error signature, and selecting a compression or readout basis; shot-count reduction is developed here as one exemplary case. Existing methods assume the symmetry is known in advance; we remove that assumption. When it is unknown, the carried symmetry is discovered from the data by a symmetry test that scores candidate groups, and the largest passing group is exploited as the measurement basis. We state the pipeline precisely, prove the selection rule is unbiased, and charge discovery in full. Two conditions are treated, both detected by the same score with a different projection: a weak condition, commutation with the representation, and a strong condition, confinement to a single charge sector, the distinction drawn in the quantum-reference-frame literature. A single circuit, a controlled twirl followed by a SWAP test, discovers both: discarding the group register tests the weak condition, post-selecting it the strong one. The framework is general over finite groups, with cyclic (Fourier), dihedral, and symmetric-group (Schur-Weyl) examples; strong confinement to the symmetric, or Dicke, subspace is an exponential reduction. Seeded demonstrations show the loop wins net of discovery: weak matching on momentum readout reduces shots by a factor widening from ten to several thousand, and strong matching on a two-system target by a further factor of the subsystem size. Blind symmetry matching is a practical primitive for the common case where the matched basis cannot be written down in advance.