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Q36 / Algorithms & primitives / question

Can local spin tests perform a spectral measurement?

Can pairwise STP approximate a spectral threshold measurement for an exchange Hamiltonian? Bound the error and resource cost, and ask separately for a protocol that retains the filtered state.

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Working mathematical formulation

For a physically specified exchange Hamiltonian H=∑eaePseH=\sum_e a_eP_s^e with polynomial input description and ∥H∥≤1\lVert H\rVert\le 1, let 0≤f≤10\le f\le 1 approximate a threshold with spectral margin Δ\Delta. Construct an STP acceptance effect EE with ∥E−f(H)∥≤ε,\lVert E-f(H)\rVert\le\varepsilon, or prove an obstruction in the stated model. Bound qubits, tests, rounds, and success probability in n,1/Δ,1/εn,1/\Delta,1/\varepsilon. Separately approximate the full spectral-filtering measurement, including the output state and its correlations with a reference.

Why this is interesting

A sequence of STP measurements defines an acceptance effect. Comparing that effect with a function of the Hamiltonian tests whether the protocol performs spectral filtering.

A place to start

Compute the effect of alternating two overlapping projectors exactly, then compare it with filters of their sum. A local-term average estimates mean energy; it is not a measurement of the Hamiltonian’s spectrum.

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

Q01 / Research directionWhich algorithmic primitives does subsystem spin suggest?Can filtering, recoupling, or state comparison give an STP algorithm for a specified task? Identify its interference mechanism and compare its resource cost with a circuit implementation.Q02 / QuestionHow can order finding be expressed through STP?Describe the encoding, modular arithmetic, and readout of an STP order-finding protocol. Identify how its multiplicity-space operations represent periodicity and interference.

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