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Q09 / Quantum resources / question

What 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.

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Working mathematical formulation

Fix NN, count all qubits in that register, and allow no fresh quantum ancillas. Start in ΠJ/Tr⁡ΠJ\Pi_J/\operatorname{Tr}\Pi_J with J=N/2−kJ=N/2-k. A uniform classical controller makes polynomially many adaptive pairwise STP measurements and decides from the outcome record with a 2/32/3 versus 1/31/3 gap. Determine its power and classical simulation cost as kk grows. The active multiplicity dimension is m(N,J)=(Nk)−(Nk−1);m(N,J)=\binom{N}{k}-\binom{N}{k-1}; separate constant kk, logarithmic kk, and extensive kk.

Scope & assumptions

The spin sector is supplied as the initial-state promise; its preparation cost is a different problem. A fixed-spin label does not specify a pure multiplicity state. J=N/2J=N/2 has multiplicity one and is inert under internal STP.

Why this is interesting

The fully symmetric sector is inert under pairwise STP, but lower sectors contain multiplicity. A useful resource theory must explain the space between these extremes.

Explicit variants

  1. Computing within a specified multiplicity space

    For a fixed total-spin sector VJ⊗MJV_J\otimes M_J, specify how a logical input is encoded in MJM_J and fix the state of VJV_J. Characterize the decision, sampling, or channel tasks realizable using adaptive STP on the closed register. Compare supplied encoded inputs with the maximally mixed sector input above, and account for any preparation or readout operations unavailable within STP.

  2. Simulation and hardness regimes

    Find structural promises on a family of multiplicity states, measurement geometries, or accessible subsystem-spin correlations that imply efficient classical simulation. Compare them with a family admitting universal STP computation under stated preparation assumptions. Seek a proved boundary in the promised structure; low total spin alone is not the promised computational resource.

A place to start

Derive exact simulation costs from the multiplicities at fixed kk, then study the first regimes in which that dimension ceases to be polynomial.

Related work

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

  1. Symmetry Protected Quantum Computation ↗

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

    Read the relevance note

Related questions

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?Q11 / QuestionDoes multiplicity-space coherence improve STP protocols?Within one total-spin sector, compare a state with its dephased version in a fixed coupling basis. Seek a separation in task success probability or measurement cost.

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