Cool stuff we should do utilizing the BlockEncoding class

[MatP1337]

Copied from #448

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The following list showcases the future functionalities we think could and should be implemented:

• Nested commutators $[A, [A, [A, ..., [A, B]$
• Custom BE from literature:
  • FABLE,
  • S-FABLE and LS-FABLE,
  • Preconditioned Block Encodings for Quantum Linear Systems,
  • Block-encoding structured matrices for data input in quantum computing,
  • Explicit Quantum Circuits for Block Encodings of Certain Sparse Matrices,
  • FOCQS-LCU, and Practical block encodings of matrix polynomials that can also be trivially controlled,
  • BE of Pauli strings using the stabilizer formalism,
  • Resource-efficient Variational Compilation of Block
    Encodings
  • Explicit Block Encodings of Discrete Laplacians with Mixed Boundary Conditions
  • Efficient and Explicit Block Encoding of Finite
    Difference Discretizations of the Laplacian.
    -> this is essentially what we've done w/ custom block encoding tuple
    • we should extend this to N-dimensions. The same permutation operator as in
  • Dictionary-based Block Encoding of Sparse Matrices with Low Subnormalization and Circuit Depth
  • Efficient block encodings for quantum chemistry simulations, e.g.:
    • Double Factorization,
    • Tensor Hypercontraction,
    • Fault-Tolerant Quantum Simulation of Materials Using Bloch Orbitals,
    • Green's functions, dipole operator,
    • Efficient Simulation of Pre-Born-Oppenheimer Dynamics on a Quantum Computer,
    • Ground State Preparation via Dynamical Cooling,
    • Quantum algorithm for vibronic dynamics: case study on singlet fission solar cell design,
    • Quantum algorithm for simulating non-adiabatic dynamics at metallic surfaces,
    • Harmonic sequence state-preparation,
    • Quantum Regularized Least Squares
    • Efficient Quantum Circuits for Non-Unitary and Unitary Diagonal Operators with Space-Time-Accuracy Trade Offs and furthermore Quantum algorithm for anisotropic diffusion and convection equations with vector norm scaling

Other interesting resources of possible applications to be considered:

• Tradeoffs between quantum and classical resources in
  linear combination of unitaries,
• Explicit block-encoding for partial differential equation-constrained optimization (utilizes LCHS, as does this paper).

[purva-thakre]

Fault-Tolerant Quantum Simulation of Materials Using Bloch Orbitals,

I am interested in this :)

[MatP1337]

Let's do it!

[MatP1337]

ref [33], Even More Efficient Quantum Computations of Chemistry Through Tensor Hypercontraction is already in the list above (as "Tensor Hypercontraction) - they provide the necessary block encodings they also used for the paper you referred to.