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Q33 / Quantum resources / question

How many swap tests does a state require?

Define swap-test count with a specified target, error, success probability, and supply of mixed qubits.

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Working mathematical formulation

Given an nn-qubit target family ρn\rho_n compatible with the permitted invariant real preparation model, define the minimum number C(ρn;ε,p,N)C(\rho_n;\varepsilon,p,N) of pair STP tests required from NN mixed qubits to prepare an output within trace distance ε\varepsilon of ρn\rho_n, conditioned on a specified set of measurement outcomes with total probability at least pp. Specify fixed versus record-dependent output registers and traced ancillas. Compare nonadaptive schedules, adaptive pair choice, and bounded-feedback protocols; count retries and fresh resources.

Why this is interesting

Preparation complexity asks how much measurement creates a particular state, rather than how hard it is to decide a language. The distinction between retaining successful pieces and restarting entire histories is essential.

A place to start

Find a natural invariant encoded family with a provable adaptive/nonadaptive separation. First test labeled coupling-tree states and repeated preparation of small resources.

Related work

These works provide context and established results. They need not be the origin of this exact formulation.

  1. The two-qubit singlet/triplet measurement is universal for quantum computing given only maximally-mixed initial states ↗

    Terry Rudolph, Shashank Soyuz Virmani · 2023

    Read the relevance note
  2. Symmetry Protected Quantum Computation ↗

    Michael H. Freedman, Matthew B. Hastings, Modjtaba Shokrian Zini · 2021

    Read the relevance note

Related questions

Q09 / QuestionWhat is the simulation cost of STP at fixed total spin?With a fixed register in sector J=N/2−kJ=N/2-k, determine how STP computation and classical simulation scale with the spin deficit kk and the multiplicity dimension.Q10 / Research directionCan antisymmetry become an operational resource?Treat measurements within fixed blocks as free and charge for measurements between blocks. Can an average monotone of singlet correlations or multiplicity information bound state-conversion cost?

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