Research questions
These are questions that arose while thinking about subsystem spin. We have tried to state the assumptions explicitly and give each question a starting point; some formulations will surely need revision.
These are working formulations, including two named literature conjectures. Sources document connections and antecedents; inclusion is not a claim of novelty or a complete prior-art audit.
Conventions for the working formulations
Unless a question says otherwise, the input is classical, the controller is uniform and runs in polynomial time, the quantum input is maximally mixed, and every pair measurement is an ideal singlet–triplet projection (STP): the measured qubits are kept, and coherence within the singlet or triplet subspace is preserved. An ancilla-based SWAP test implements this measurement. All ancillary qubits and failed attempts count toward the stated cost. A closed-register statement permits no fresh qubits or discarding.
Decision protocols use a versus acceptance gap. Sampling error means total variation distance; state-preparation error means trace distance. Coherent encoded-channel simulation is stronger and uses diamond distance, including an external reference.
Which 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.
How 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.
How efficiently can we compile into STP?
How much of the cost of a universal STP construction is essential?
Can STP implement quantum walks on representation graphs?
Specify a graph of coupling paths or irrep labels and an associated quantum walk. Determine the cost of implementing its transitions or reflections using STP.
Which Schur sampling and transform tasks can STP implement?
Determine what representation data random pair measurements reveal, when adaptivity improves estimation, and the cost of retaining coherent Schur information.
What can constant- and logarithmic-depth STP compute?
For constant or logarithmic STP depth, characterize decision power and classical sampling complexity under stated restrictions on the classical controller.
How does the number of adaptive rounds affect STP?
Compare nonadaptive, constant-round, and polynomial-round STP for decision and sampling, keeping the classical controller and error conventions fixed. No postselection is allowed.
How does interaction geometry constrain STP?
Restrict pair measurements to an interaction graph. Characterize classical simulation, universal computation, and routing overhead on paths, trees, and lattices.
What is the simulation cost of STP at fixed total spin?
With a fixed register in sector , determine how STP computation and classical simulation scale with the spin deficit and the multiplicity dimension.
Can 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?
Does 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.
Which subsystem-spin profiles can we reach?
Which patterns of pair and subsystem spin are compatible, and which can be created efficiently?
The singlet conjecture
For an exchange-symmetric interaction, does subtracting a triplet projector ever increase the complexity of its positive-weight local-Hamiltonian problem?
Is the problem in ?
has an efficient quantum algorithm through Lee–Yang theory. Does it also have an efficient classical algorithm?
What is the STP cost of preparing a Quantum Max-Cut witness?
Compare verification by a weighted random-edge singlet test with STP preparation of a state whose Quantum Max-Cut objective is near optimal.
How are product tests expressed in subsystem spin?
Express the product test as triplet projections on corresponding pairs from two unentangled registers. Analyze completeness and soundness when it is used in a two-witness verification protocol.
Can purification give a representation-theoretic adversary method?
Purify an oracle ensemble into an inaccessible register, decompose it into representation sectors, and bound the change of a progress observable per query.
Can spin structure explain disentangler bounds?
For approximate disentanglers, relate input dimension to the requirements that outputs be nearly separable and that every separable target be approximately attainable.
Could be contained in ?
For each YES instance, can a classical cut and a polynomial-size interpolation recipe support efficient quantum preparation of a low-energy state? Account for the magnetization sector and spectral gap.
Which computational tasks does Schur–Weyl duality relate?
Compare permutation-based and STP models with matched input and output conventions, and identify the additional resources needed for efficient reductions.
Which representation–extraction tasks can STP implement?
Determine which coherent operations in a representation–extraction duality are available from STP, and whether they yield verification or state-generation tasks.
Which spin problems are -complete?
Formulate promise problems on spin networks, recoupling structures, or adaptive measurement histories, with explicit reductions and precision requirements.
What is an -symmetric oracle problem?
Define invariant unitary or measurement oracles and compare query costs under explicit access conventions.
What changes with higher-arity total-spin measurements?
Compare the computational power, depth, and adaptivity of pairwise STP with measurements on larger subsets, including the cost of reproducing both their outcomes and the states they leave afterward.
Which irrep measurements are useful for qudits?
For larger local dimension, exchange symmetry and physical spin no longer identify every sector in the same way.
Can spin protocols be translated into bosonic or fermionic mode models?
Construct explicit encodings and compare preparation and measurement costs under specified particle-number and mode constraints.
How accurately can physical devices implement singlet–triplet projection?
Compare physical singlet/triplet measurements with the ideal projection, including coherence within each sector, leakage, loss, and repeatability.
Which errors does an STP encoding protect against?
Analyze protection and fault-tolerance overhead under specified collective noise, noncollective noise, measurement errors, and leakage.
Can subsystem spin give a common description of quantum computation?
The guiding hypothesis concerns a shared explanation of interference, entanglement, walks, transforms, and measurement-based computation.
What limits STP preparation of accepted quantum witnesses?
Bound the STP cost of preparing a state close to an accepted witness, with explicit trace-distance error and success probability.
How can a symmetric block acquire lower subsystem spin?
Internal STP leaves a fully symmetric register unchanged. Which added systems and interactions let a preparation process leave that limit?
Can intermediate state resources guide an STP compiler?
Schedule a finite library of STP state conversions, reuse states from unsuccessful attempts, and bound the resulting compilation cost.
How many swap tests does a state require?
Define swap-test count with a specified target, error, success probability, and supply of mixed qubits.
How hard is predicting singlet transmission under STP?
Use a final singlet correlation with an untouched reference as the output of a computation.
Quantum spin-SAT: which subsystem spins can be forbidden?
Each constraint forbids a specified total spin on a specified subsystem. Can one quantum state satisfy all the constraints?
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.
What changes with exponentially many measurements?
Classify exponentially long STP protocols with polynomial quantum workspace and explicit classical controller-space bounds.
What is the Quantum Max-Cut optimum at fixed total spin?
Bound the fixed-spin optimum and the energy lost by restricting states to specified cluster or dimer ansätze.
Which compact-group irrep measurements support universal computation?
Replace qubits and by a fixed representation of a compact group. Classify the computation generated by measuring irreducible labels on small subsets.
Which spin-constraint problems are naturally -complete?
Can a classical sequence of pairings, recouplings, or an adiabatic path certify a spin-constraint problem? Identify natural complete problems with explicit verification and promise gaps.
What is an analogue of Grover search?
What should “unstructured search” mean when the available operations are subsystem-spin measurements? Is there a direct quadratic speedup over classical search, and what coherent or adaptive structure produces it?
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