Prepare an input state

Why this is a layer

Dense unstructured preparation costs $\Theta(2^n)$ gates, and ancillas do not remove it. Yuan and Zhang pin the depth to $\Theta(n + 2^n/(n+m))$ and the size to $\Theta(2^n)$ for any number $m$ of ancillary qubits. Ancillas buy depth, not gates. So a pipeline that loads a dense unstructured right-hand side pays $\Omega(N)$ gates at the input and has no exponential advantage regardless of how good the solver above it is. An end-to-end exponential speedup requires $b$ to be sparse, analytically specified, low-bond-dimension, or produced by an earlier quantum subroutine — never that it be an arbitrary classical vector. That description-dependent cliff, not the solver, is what decides the pipeline.

Records for what it handles

The catalogue's own entries for the objects on either side of this contract.

Ways to do this

3 methods recorded

• Uniformly controlled rotations

  Prepare an arbitrary state with one layer of uniformly controlled (multiplexed) Ry and Rz rotations per qubit, the angles computed analytically from the amplitude list. This is the exact, assumption-free method most software stacks emit by default.

• Grover-Rudolph bisection preparation

  Prepare a discrete approximation to a probability density by recursive bisection: at layer $k$ a uniformly controlled rotation splits each current interval's probability mass between its two halves, so only $n$ rotation layers are needed.

• Sparse state preparation

  When only $d$ of the $2^n$ amplitudes are nonzero, build the $d$ computational-basis strings directly instead of rotating through the whole binary tree, so the cost tracks $d$ and $n$ rather than $2^n$.

Routes that skip this layer

No recorded route avoids this step.

This is a step inside

• HHL: eigenvalue inversion by phase estimation

  Prepare $|b>$, run phase estimation against $e^{-iAt}$ to write eigenvalue estimates into an ancilla register, apply a controlled rotation with amplitude proportional to $1/λ̃$, uncompute the estimation and post-select on the rotation ancilla. The success amplitude is about $1/κ$, so the procedure is amplified $O(κ)$ times.

• QSVT matrix inversion

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

• Chebyshev LCU inversion

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

• Discrete adiabatic inversion

  Encode the solution as the null eigenstate of a Hamiltonian path built from $A$ and $|b>$, 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 $\log(κ)$ 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 $|x>$ is the null eigenstate of a Hamiltonian built from $A$ and $|b>$, so one application of the filter solves it once a starting state with non-trivial overlap is supplied.

• Block-encoding from a Pauli decomposition

  Write $A = Σ_j c_j P_j$ over Pauli strings; PREPARE loads amplitudes proportional to $sqrt(|c_j|)$ into an ancilla register, SELECT applies the controlled Pauli strings, and PREPARE$†$ unprepares, leaving $A/||c||_1$ in the block flagged by the all-zeros ancilla. This is the input model chemistry and lattice Hamiltonians supply for free.

• Fixed-point amplification

  Replace the $±1$ reflections of textbook amplification with tuned phase shifts, so the iteration converges on the target instead of rotating past it — which is what happens when the initial success amplitude is known only as a lower bound.

• Truncated-Taylor LCU simulation

  Truncate the Taylor series of $e^{-iHt}$ over short segments and implement the truncated sum as a linear combination of unitaries — PREPARE loads the coefficients, SELECT applies the terms, PREPARE$†$ unprepares — with the all-zeros ancilla flag amplified.

• Direct sampling in a measurement basis

  Decompose $O$ into Pauli strings, rotate each into the computational basis with a layer of single-qubit Cliffords, sample bitstrings, and recombine the per-term averages linearly. No ancilla, no controlled operations, minimum added depth.

• Coherent amplitude-estimation readout

  Encode the expectation value into an amplitude and estimate that amplitude coherently — phase estimation on the Grover operator $Q = −A S_0 A^{-1} S_χ$, or one of the QPE-free variants — instead of averaging independent shots.

• Classical shadow readout

  Apply a random unitary from a chosen ensemble, measure in the computational basis, and keep the (unitary, outcome) pair; inverting the measurement channel turns each pair into an unbiased single-shot snapshot of $ρ$, and median-of-means over snapshots predicts many observables at once. The observables may be chosen after the data has been taken.

• Measure commuting terms together

  A Hamiltonian's terms are measured one group at a time rather than one term at a time. Terms that commute qubit-wise can share a single set of measurements, so the question becomes how few groups the terms can be covered by — a graph problem, and a hard one.

• Ground-state energy by phase estimation

  Prepare a state that already overlaps the ground state, evolve it under the molecular Hamiltonian, and read the energy off the accumulated phase. Nothing is optimized and nothing is varied — the answer is a measured eigenvalue, and the whole difficulty moves into the starting state.

• Tensor hypercontraction block encoding

  Factorize the chemistry Hamiltonian's two-electron integrals into a product of much smaller matrices first, then build the block-encoding of the factorized form. The saving is not in the encoding technique but in what is being encoded — a tensor with far fewer independent entries than the one the basis handed you.

Routes that run this slot many times

For these routes this slot is inside a loop, so its cost is multiplied rather than paid once. That multiplier is usually the largest single term in what the route costs.

• HHL: eigenvalue inversion by phase estimation runs O(κ) times — once per amplification round

  The loop stays coherent: nothing is measured between turns. The preparation may still be reapplied every turn — what the loop never pays is a readout and a restart from classical data. The price is depth, and a success probability that multiplies down the chain.

• Direct sampling in a measurement basis runs O(1/ε²) shots, and one preparation per shot

  The loop closes through a measurement: every turn ends in a readout and starts from a fresh preparation. The price is a count of runs, not a depth.

• Coherent amplitude-estimation readout runs M iterations, M = O(1/ε), each running the preparation forwards and backwards once

  The loop stays coherent: nothing is measured between turns. The preparation may still be reapplied every turn — what the loop never pays is a readout and a restart from classical data. The price is depth, and a success probability that multiplies down the chain.

• Classical shadow readout runs N = O(log(M) · max_i ||·||²_shadow / ε²) measurements, one preparation each

  The loop closes through a measurement: every turn ends in a readout and starts from a fresh preparation. The price is a count of runs, not a depth.

No multiplicity recorded

These routes take this step and no source we have read says how often. That is an absence, not a claim that they take it once.

• QSVT matrix inversion

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

• Chebyshev LCU inversion

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

• Discrete adiabatic inversion

  Encode the solution as the null eigenstate of a Hamiltonian path built from $A$ and $|b>$, 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 $\log(κ)$ 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 $|x>$ is the null eigenstate of a Hamiltonian built from $A$ and $|b>$, so one application of the filter solves it once a starting state with non-trivial overlap is supplied.

• Block-encoding from a Pauli decomposition

  Write $A = Σ_j c_j P_j$ over Pauli strings; PREPARE loads amplitudes proportional to $sqrt(|c_j|)$ into an ancilla register, SELECT applies the controlled Pauli strings, and PREPARE$†$ unprepares, leaving $A/||c||_1$ in the block flagged by the all-zeros ancilla. This is the input model chemistry and lattice Hamiltonians supply for free.

• Fixed-point amplification

  Replace the $±1$ reflections of textbook amplification with tuned phase shifts, so the iteration converges on the target instead of rotating past it — which is what happens when the initial success amplitude is known only as a lower bound.

• Truncated-Taylor LCU simulation

  Truncate the Taylor series of $e^{-iHt}$ over short segments and implement the truncated sum as a linear combination of unitaries — PREPARE loads the coefficients, SELECT applies the terms, PREPARE$†$ unprepares — with the all-zeros ancilla flag amplified.

• Measure commuting terms together

  A Hamiltonian's terms are measured one group at a time rather than one term at a time. Terms that commute qubit-wise can share a single set of measurements, so the question becomes how few groups the terms can be covered by — a graph problem, and a hard one.

• Ground-state energy by phase estimation

  Prepare a state that already overlaps the ground state, evolve it under the molecular Hamiltonian, and read the energy off the accumulated phase. Nothing is optimized and nothing is varied — the answer is a measured eigenvalue, and the whole difficulty moves into the starting state.

• Tensor hypercontraction block encoding

  Factorize the chemistry Hamiltonian's two-electron integrals into a product of much smaller matrices first, then build the block-encoding of the factorized form. The saving is not in the encoding technique but in what is being encoded — a tensor with far fewer independent entries than the one the basis handed you.

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.