Strong empiricalBlock encoding · LCU
Linear combination of unitaries
A block-encoding primitive that turns a weighted sum of unitary operations into one larger unitary circuit.
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The defining behavior was checked exactly: a mathematical identity, a full statevector or stabilizer simulation, or an exhaustive basis-state truth table.
The design was verified by construction plus measured evidence: statistical re-execution, small-instance analytic agreement, sub-block, echo, or invariant checks. Scale-specific bugs can still survive.
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A block-encoding primitive that turns a weighted sum of unitary operations into one larger unitary circuit.
A single-qubit rotation sequence that transforms an encoded signal into a polynomial response.
Given a function f and a positive semi-definite matrix A whose eigenvalues are λⱼ, estimate the spectral sum Tr[f(A)] = Σⱼ f(λⱼ), a family whose typical examples the paper gives as the von Neumann entropy, the trace of A⁻¹, the log-determinant and the Schatten p-norm, the last of which it says does not require the matrix to be positive semi-definite.
Represent the spectrum of a quantum chemistry Hamiltonian, given in an arbitrary (for example molecular) orbital basis, as a block-encoded quantum circuit cheap enough to support phase estimation of a molecular eigenvalue.
A linear-algebra reference that forces the catalog to show input loading, conditioning, and output observability.
Adapt the Harrow-Hassidim-Lloyd (HHL) quantum linear-systems algorithm, most of whose components current noisy quantum hardware cannot reach, into a form that near-term devices can actually execute, and demonstrate it on an application.
Encode a linear kinetic plasma problem — modeling electrostatic linear waves, driven by a spatially localized external current, in a one-dimensional Maxwellian electron plasma — into a quantum circuit that solves the resulting linear system.
Given quantum access to a historical record of asset returns, determine the optimal risk-return tradeoff curve of a portfolio and provide a way to sample from the optimal portfolio.
A framework for applying bounded polynomial transformations to singular values of a block-encoded matrix.
Given query access to the entries of a d × d Hermitian matrix A, output a classical description of a good approximation of its top eigenvector, the eigenvector belonging to the largest eigenvalue.
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