Estimate an observable

Why this is a layer

Reading out the full solution vector destroys the speedup. A classical description of an $n$-qubit state is $2^n$ numbers before any sampling cost is counted, and reconstructing it costs $O(rank(ρ)·d/ε²) \leq O(d²/ε²)$ copies for trace-distance error $\varepsilon$ with $d = 2^n$ (O'Donnell and Wright — upper bounds only). This layer exists because the useful question is almost always a scalar: one $⟨x|M|x⟩$ rather than $x$ itself. That is what HHL actually delivers, and it is one of the conditions Aaronson's "Read the fine print" (Nature Physics 11, 291–293, 2015) names as carrying the exponential claim. The number of runs is decided here and decided independently of the preparation above it: the same |ψ⟩ costs ε^-2 runs by plain sampling and ε^-1 by coherent estimation, a difference that can exceed the advantage being claimed upstream.

Records for what it handles

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

Ways to do this

4 methods recorded

• 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.

Routes that skip this layer

No recorded route avoids this step.

This is a step inside

• Quantum simulation of the KvN representation

  Because the Koopman-von Neumann generator is Hermitian and its propagator unitary, the lifted evolution can be run by Hamiltonian simulation directly. No linear system is assembled and no linear solver is called.

• Variational quantum eigensolver

  Prepare a parameterised trial state on the quantum computer, measure the Hamiltonian's expectation value in it, and let a classical optimiser move the parameters. The quantum computer never runs a long coherent evolution; it runs a short one many times, and the loop closes through a classical number.

• Variational imaginary-time evolution

  Keep the parameterised trial state, but stop treating the parameters as something to optimise: derive their equation of motion from a variational principle and integrate it in imaginary time. The parameters move because a differential equation says where they go, not because a search tried somewhere and liked the answer.

• ADAPT-VQE adaptive ansatz

  Refuse to fix the structure in advance. Start from nothing and add one operator at a time, choosing each from a pool according to what the molecule itself indicates, until the energy stops improving. The circuit ends up short because nothing was included that the problem did not ask for.

• qubit-ADAPT-VQE ansatz a narrower version of ADAPT-VQE adaptive ansatz

  The same grow-it-one-operator-at-a-time construction, with the pool rebuilt out of qubit operators rather than fermionic excitations so that the circuits it produces are shallow enough for near-term hardware.

• Qubit coupled-cluster ansatz

  Skip the fermionic layer and build the ansatz directly in qubit space, ranking candidate entangling operators by how much each would move the energy and keeping the ones that earn their place.

• Analytic-gradient parameter search

  Get the gradient of the objective exactly, rather than by finite differences, by running the same circuit again at shifted parameter values. The direction is then not an estimate of a slope taken from two noisy numbers; it is the slope, estimated to whatever precision the shots allow.

• Variational quantum deflation

  Find the ground state first, then run the same variational search again with a term that punishes overlap with every state already found. Each state is reached by pushing the search off the ones below it, so they have to be found in order and each one costs another pass through the whole loop.

• Subspace-search variational eigensolver

  Send several mutually orthogonal input states through one parameterised circuit and minimise their energies together. A unitary keeps orthogonal inputs orthogonal, so the whole low-energy subspace comes out of a single optimisation — no earlier state to deflate, and no ancilla to test overlaps with.

• Quantum subspace expansion

  Stop optimising and start measuring: take the state the ground-state run already produced, measure matrix elements in a small space of operators applied to it, and let a classical solver diagonalise the little matrix that results. The extra accuracy is bought with classical work and more measurements, not with more coherence.

• Quantum equation of motion

  Ask for the gaps directly rather than for two energies to subtract. Measure the matrix elements the classical equation-of-motion formalism needs between excitation operators on the ground state, and solve its generalised eigenvalue problem classically; what comes back is an excitation energy, which is the quantity a spectrum is made of.

• Folded-spectrum variational eigensolver

  Point the same search somewhere other than the bottom. Minimising the variance around a chosen energy makes every eigenstate a minimum and the one nearest that energy the reachable one — so a state can be found without knowing its index, and the bill arrives as a squared Hamiltonian with far more terms to measure.

• Penalty-constrained variational eigensolver

  Add a term to the objective that punishes the trial state for leaving the symmetry sector you asked for, and the ordinary ground-state search returns that sector's lowest state — an excited state of the whole Hamiltonian whenever the sector is not the one the ground state lives in. Which penalty is used matters: one common form is proved not to work.

• Multistate contracted variational eigensolver

  Optimise one circuit for several states at once and read off the transitions between them — both the energy of each transition and the oscillator strength that says how strongly light drives it. The answer is a spectrum with intensities, which is what an absorption experiment actually produces.

• TETRIS-ADAPT-VQE ansatz a narrower version of ADAPT-VQE adaptive ansatz

  Keep ADAPT's habit of growing the ansatz from measured gradients, and stop adding one operator per round. Several operators acting on disjoint qubits can go in together, filling the same layer instead of stacking — the same circuit, packed rather than piled.

• Iterative qubit coupled cluster a narrower version of Qubit coupled-cluster ansatz

  Stop growing the circuit and grow the Hamiltonian instead. Each round folds the entanglers found so far into the operator by a canonical transformation, so every round runs a circuit of the same size — the cost moves off the device and into the number of terms that have to be measured.

• Batched ADAPT-VQE ansatz a narrower version of ADAPT-VQE adaptive ansatz

  Keep ADAPT's habit of growing the ansatz from measured gradients, and stop adding exactly one operator per round. Every operator whose gradient is close to the largest goes in together, so the ansatz reaches the same size in far fewer rounds — and it is the rounds, not the operators, that cost measurements.

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.

• Variational quantum eigensolver runs Once per iteration of the classical optimiser, and nothing bounds the iteration count — it is measured on the instance, never proved. Each of those evaluations is itself a pass over the Hamiltonian's terms, every term estimated to additive error by repeated preparation and measurement, so the shot budget multiplies through two nested counts.

  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.

• Subspace-search variational eigensolver runs Once per input state in the subspace, per optimiser iteration. The paper's construction is a search over a subspace spanned by several orthogonal inputs, so the objective is a function of all of their energies and one evaluation of it is one energy estimate for each — the price of not needing the lower states in advance. The number of inputs is k+1 for the k-th excited state; nothing bounds the iteration count, for the same reason nothing bounds VQE's.

  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.

• Quantum simulation of the KvN representation

  Because the Koopman-von Neumann generator is Hermitian and its propagator unitary, the lifted evolution can be run by Hamiltonian simulation directly. No linear system is assembled and no linear solver is called.

• Variational imaginary-time evolution

  Keep the parameterised trial state, but stop treating the parameters as something to optimise: derive their equation of motion from a variational principle and integrate it in imaginary time. The parameters move because a differential equation says where they go, not because a search tried somewhere and liked the answer.

• ADAPT-VQE adaptive ansatz

  Refuse to fix the structure in advance. Start from nothing and add one operator at a time, choosing each from a pool according to what the molecule itself indicates, until the energy stops improving. The circuit ends up short because nothing was included that the problem did not ask for.

• qubit-ADAPT-VQE ansatz a narrower version of ADAPT-VQE adaptive ansatz

  The same grow-it-one-operator-at-a-time construction, with the pool rebuilt out of qubit operators rather than fermionic excitations so that the circuits it produces are shallow enough for near-term hardware.

• Qubit coupled-cluster ansatz

  Skip the fermionic layer and build the ansatz directly in qubit space, ranking candidate entangling operators by how much each would move the energy and keeping the ones that earn their place.

• Analytic-gradient parameter search

  Get the gradient of the objective exactly, rather than by finite differences, by running the same circuit again at shifted parameter values. The direction is then not an estimate of a slope taken from two noisy numbers; it is the slope, estimated to whatever precision the shots allow.

• Variational quantum deflation

  Find the ground state first, then run the same variational search again with a term that punishes overlap with every state already found. Each state is reached by pushing the search off the ones below it, so they have to be found in order and each one costs another pass through the whole loop.

• Quantum subspace expansion

  Stop optimising and start measuring: take the state the ground-state run already produced, measure matrix elements in a small space of operators applied to it, and let a classical solver diagonalise the little matrix that results. The extra accuracy is bought with classical work and more measurements, not with more coherence.

• Quantum equation of motion

  Ask for the gaps directly rather than for two energies to subtract. Measure the matrix elements the classical equation-of-motion formalism needs between excitation operators on the ground state, and solve its generalised eigenvalue problem classically; what comes back is an excitation energy, which is the quantity a spectrum is made of.

• Folded-spectrum variational eigensolver

  Point the same search somewhere other than the bottom. Minimising the variance around a chosen energy makes every eigenstate a minimum and the one nearest that energy the reachable one — so a state can be found without knowing its index, and the bill arrives as a squared Hamiltonian with far more terms to measure.

• Penalty-constrained variational eigensolver

  Add a term to the objective that punishes the trial state for leaving the symmetry sector you asked for, and the ordinary ground-state search returns that sector's lowest state — an excited state of the whole Hamiltonian whenever the sector is not the one the ground state lives in. Which penalty is used matters: one common form is proved not to work.

• Multistate contracted variational eigensolver

  Optimise one circuit for several states at once and read off the transitions between them — both the energy of each transition and the oscillator strength that says how strongly light drives it. The answer is a spectrum with intensities, which is what an absorption experiment actually produces.

• TETRIS-ADAPT-VQE ansatz a narrower version of ADAPT-VQE adaptive ansatz

  Keep ADAPT's habit of growing the ansatz from measured gradients, and stop adding one operator per round. Several operators acting on disjoint qubits can go in together, filling the same layer instead of stacking — the same circuit, packed rather than piled.

• Iterative qubit coupled cluster a narrower version of Qubit coupled-cluster ansatz

  Stop growing the circuit and grow the Hamiltonian instead. Each round folds the entanglers found so far into the operator by a canonical transformation, so every round runs a circuit of the same size — the cost moves off the device and into the number of terms that have to be measured.

• Batched ADAPT-VQE ansatz a narrower version of ADAPT-VQE adaptive ansatz

  Keep ADAPT's habit of growing the ansatz from measured gradients, and stop adding exactly one operator per round. Every operator whose gradient is close to the largest goes in together, so the ansatz reaches the same size in far fewer rounds — and it is the rounds, not the operators, that cost measurements.

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.