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SlotLayer 2

Matrix function

Given a block-encoding of AA and a target function ff bounded on [−1,1][-1,1], produce a circuit whose designated block is an ε\varepsilon-approximation of ff applied to the singular values (or eigenvalues) of AA.

Takes

A block-encoding of AA together with its subnormalisation α\alpha; a target function ff on [−1,1][-1,1]; an error budget ε\varepsilon.

Returns

A circuit implementing a block-encoding of f(A)f(A) to error ε\varepsilon, together with the query count in UU and U†U^\dagger.

This one, drawn

Matrix functionQuantum singular value transformation · 2 parts insideQuantum singular value transformation · 2 parts inside — click the line to open it hereChebyshev series by linear combination of unitaries · 2 parts insideChebyshev series by linear combination of unitaries · 2 parts inside — click the line to open it hereQuantum singular value transformation · 2 parts inside — click the name to read about itQSVTChebyshev series by linear combination of unitaries · 2 parts inside — click the name to read about itChebyshev series by LCUBlock-encoding — you start hereBlock-encoding of f(A) — you finish here

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From Block-encoding to Block-encoding of f(A)

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Why this is a layer

Inverting AA is one choice of ff; the same machinery gives Hamiltonian simulation, spectral filtering and thresholding. The alternatives here differ materially: QSVT compiles ff into a phase sequence and needs one ancilla qubit, an LCU of Chebyshev terms needs no phase factors but pays in ancillas and in the coefficient sum, and where the matrix is close to low rank and sampling access is granted, a classical algorithm does the same job in time independent of the dimension.

Ways to do this

2 methods recorded

  • Quantum singular value transformation

    Interleave the block-encoding UU, its inverse, and projector-controlled phase shifts eiφ(2Π−I)e^{iφ(2Π-I)} so that the designated block becomes PP applied to the singular values of AA. The phase sequence is the compiled form of the polynomial, and a single ancilla qubit carries the phase shifts.

  • Chebyshev series by linear combination of unitaries

    Write the target function as a Chebyshev series, then implement that series as a linear combination of walk-operator powers using a PREPARE/SELECT pair on an ancilla register, and post-select. No phase factors are computed — the polynomial enters through the coefficients of the combination instead.

Routes that skip this layer

No recorded route avoids this step.

This is a step inside

  • QSVT matrix inversion

    Block-encode AA, apply the quantum singular value transformation with an odd polynomial approximating a scaled 1/x1/x away from the origin, then amplify. Because it acts on singular values, AA need not be Hermitian or sparse — only block-encodable.

  • Chebyshev LCU inversion

    Expand an approximation of 1/x1/x in Chebyshev polynomials and implement that expansion directly through the quantum walk associated with a sparse AA, rather than through Hamiltonian simulation. Phase estimation is never used, which is what removes the poly(1/ε)\mathrm{poly}(1/\varepsilon) bottleneck.

  • Discrete adiabatic inversion

    Encode the solution as the null eigenstate of a Hamiltonian path built from AA and ∣b>|b>, follow that path with a sequence of qubitization walk operators to fixed precision, then finish with an eigenstate filter implemented as a linear combination of walk operators rather than by quantum signal processing. Costa and co-authors prove an adiabatic theorem for intrinsically discrete-time evolutions, which removes the residual log⁡(κ)\log(κ) that continuous adiabatic treatments carried.

  • Eigenstate filtering inversion

    Construct the minimax-optimal polynomial that is 1 at a target eigenvalue and uniformly small outside a spectral gap, and apply it through quantum signal processing. For a linear system ∣x>|x> is the null eigenstate of a Hamiltonian built from AA and ∣b>|b>, so one application of the filter solves it once a starting state with non-trivial overlap is supplied.

In the Atlas

No record in the Atlas covers this yet. The catalogue is circuits and primitives; this part of the literature is not in it.