MethodLayer 1
Hardware-efficient ansatz
Build the trial state out of the gates and couplings the machine already has, and accept whatever states that reaches. The circuit is shallow because nothing in it was chosen for the chemistry; there is correspondingly no argument that the state you want is inside the family.
The Hamiltonian whose ground state is wanted, together with whatever structure is to be respected — particle number, spin, point-group symmetry, a reference determinant — and the connectivity and native gate set of the device the family has to run on.
A circuit family with a fixed structure and free real parameters, together with the number of those parameters — which is the size of the classical search problem handed to the next layer.
Same contract as the slot it fills.
This one, drawn
From Hamiltonian whose eigenvalues are wanted to Parameterised circuit family
A circle is an object you are holding. This method is drawn heavier, opened into its own steps; the other lines between the same two ends are the alternatives recorded for the same slot. Circles are named on hover, and each one is a link.
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What it fills
- Choose a parameterised trial state
Fix the gate structure of a circuit family and leave its rotation angles open. What comes back is not a circuit but the set of states the later optimisation is allowed to search — which is why this is a slot of its own and not a paragraph in one method's write-up.
When it applies
Kandala et al. describe the family as "trial states specifically tailored to the available interactions in our quantum processor" — which is the definition and also the limitation, since the family is then a property of that processor rather than of the problem. Their demonstration is bounded and they state its bounds: up to six qubits, Hamiltonians "with over a hundred Pauli terms", molecules up to BeH2, and it required "a compact encoding of fermionic Hamiltonians and a robust stochastic optimization routine" alongside the ansatz itself.
Requires
Every step this method names moves its route along, so there is nothing it needs alongside them.
Example
given N # qubit count, fixed by the encoding of the target Hamiltonian H
# upstream of this ansatz -- nothing else about H enters below
d # trial-state depth; a design choice, not computed from H --
# Algorithm 1 fixes d by hand and only increases it while E_f
# has not converged (Sec. II, Algorithm 1, steps 2, 6, 14)
U_ENT = exp(-i H_0 tau)
# generated by the processor's OWN drift Hamiltonian H_0 and
# evolution time tau (p. 2, before Eq. 2) -- both are
# controllable, set once from simulation to a concurrence-
# maximizing phase, then held fixed through the optimization,
# not chosen for, or derived from, H (p. 2-3)
requires U_ENT generates "sufficient entanglement"
# no specific two-qubit gate is assumed or required to be implemented accurately;
# contrast UCC, which needs high-fidelity gates approximating a unitary tailored
# to a theoretical ansatz (p. 3)
# the experiment happens to realize U_ENT as a sequence of cross-resonance gates,
# but that choice is not part of the construction (p. 3)
# numerical simulations found accurate optimization already holds with U_ENT fixed
# (fixed tau, fixed H_0 couplings) -- so only the Euler angles below are varied,
# not U_ENT itself (p. 3)
# --- ansatz-construction: alternate single-qubit Euler layers with the fixed entangler --
|Phi(theta)> <- |0>^(x)N # N-qubit reference state
for i = 0 to d:
for q = 1 to N:
if i == 0:
apply X(theta[q,0,2]) Z(theta[q,0,3]) to qubit q
# the first Z rotation of U(q,0) is not implemented, since the
# qubits start in |0> (p. 2, remark following Eq. 2)
else:
apply Z(theta[q,i,1]) X(theta[q,i,2]) Z(theta[q,i,3]) to qubit q
# = U(q,i)(theta), an arbitrary single-qubit Euler rotation built
# from Z and X gates (Fig. 1c, Eq. 2)
if i < d:
apply U_ENT to all N qubits
# entangles all the qubits in the circuit, not only neighboring pairs
# (p. 2)
return |Phi(theta)>
# the parameterized trial-state family handed to the energy-estimation and
# classical-optimization steps that follow (Sec. II, Algorithm 1)Cost, as the source states it
Kandala et al.: variational angles for qubits and entanglers — for the six-qubit BeH run at — and two energy estimates per gradient step, "regardless of the number of variables", by SPSA ( samples each; for the final energy, averaged over the last 25 updates). Sampling error is for samples per Pauli term (Eq. 14), over 4, 99 and 164 terms in 2, 25 and 44 TPB sets for H, LiH and BeH. No gate count in is given, and nothing bounds the iterations: Algorithm 1 runs "while has not converged", with , and the sample counts chosen by hand. In place of a bound, simulations (ideal gates, two angles per layer, no decoherence, 10 runs) reach chemical accuracy at depths , while decoherence and finite sampling "set the optimal depth for optimizations on our quantum hardware to entanglers"; Section VII.C measures a depolarizing noise rate of as what chemical accuracy needs at 6 and 8 entanglers on four-qubit LiH.
Implementations
IBM transmon VQE: H2, LiH, BeH2
The experiment the ansatz was introduced for. Kandala et al. run a variational eigensolver on an IBM superconducting quantum processor, reporting in the abstract "the experimental optimization of up to six-qubit Hamiltonian problems with over a hundred Pauli terms, determining the ground state energy for molecules of increasing size, up to BeH2." The chip carries seven transmons — six fixed-frequency plus a central weakly-tunable asymmetric transmon — of which six are used, the central tunable qubit among them. Potential energy surfaces are measured for H2 on 2 qubits, LiH on 4 qubits and BeH2 on 6 qubits, all at depth d = 1, over a range of interatomic distances.
Trial states alternate layers of single-qubit Euler rotations with an entangler generated by the processor's own drift Hamiltonian, realised here as a sequence of two-qubit cross-resonance gates. Hamiltonian tomography of those gates "is used to reveal the strengths of the various interaction terms, and the gate time for maximal entanglement" — but the gates are deliberately not run at that point: the two-qubit gate time is set to 150 ns, chosen so that the corresponding gate phases "lie at the beginning of the minimal error plateaus, in order to minimize the effect of decoherence while delivering sufficient entanglement." The entangler evolution time tau is therefore fixed rather than varied, at the start of a plateau of minimal energy error that simulation locates near — but not at — maximal pairwise concurrence. Z rotations are frame changes in the control software and X rotations are amplitude-scaled calibrated X_pi pulses, at a fixed total time of 100 ns per single-qubit rotation. Energies come from correlating single-qubit Z-basis dispersive readouts after post-rotations, with the Pauli terms grouped into tensor product basis sets that share post-rotations; readout assignment errors are handled by running a readout calibration before sampling at every angle update and correcting the outcomes with it. The classical loop is SPSA, which approximates the gradient from two energy evaluations per step regardless of the number of parameters p; c = 10^-1 is used in all the experiments, and a is calibrated from an energy difference averaged over 25 random gradient directions. Alongside the depth-1 potential energy surfaces, a separate experimental optimization of the 4-qubit LiH Hamiltonian at its bond distance is run at d = 0, 1 and 2 (Fig. S9a), one experimental run per depth. The device is cooled in a dilution refrigerator anchored at 25 mK; across the six qubits Table S1 lists T1 from 20.4 +/- 4.4 to 44.4 +/- 4.9 microseconds and readout assignment errors from 0.0240 to 0.0544.
Molecular Hamiltonians in the STO-3G basis, with one- and two-electron integrals obtained from the PyQuante software. H2 has 4 spin-orbitals (the spin-degenerate 1s orbitals of the two hydrogens), mapped by a binary tree encoding onto 4 qubits and reduced to 2 by removing the two spin-parity qubits. LiH and BeH2 take 1s on each H plus 1s, 2s and 2px on the heavy atom, for 8 and 10 spin-orbitals respectively; the heavy atom's two innermost 1s spin-orbitals are then assumed perfectly filled after dressing via diagonalization of the non-interacting part of the fermionic Hamiltonian, and the resulting 8 spin-orbital BeH2 Hamiltonian is parity-mapped and reduced to 6 qubits, with LiH handled the same way onto 4. After tapering the Hamiltonians carry 4, 99 and 164 Pauli terms on 2, 4 and 6 qubits, in 2, 25 and 44 tensor product basis sets, and are listed term by term at the bond distance in Table S2. The detailed optimization in Fig. 2 is BeH2 at interatomic distance l = 1.7 Angstrom; the separate depth comparison of Fig. S9a takes the 4-qubit LiH Hamiltonian at its bond distance, which that figure's panel labels 1.545 Angstrom.
The six-qubit BeH2 optimization varies 30 Euler control angles simultaneously (p = N(3d+2) at N = 6, d = 1), takes each gradient from 10^3 samples per energy estimate at the two perturbed angle sets, and reports a final energy from the angles averaged over the last 25 updates, re-estimated with a higher 10^5 samples. For all three molecules the experimental points are plotted against exact diagonalization and against density plots of 100 noisy numerical optimizations per interatomic distance; the experimental error bars are smaller than the markers, and the paper's stated result is agreement with its noise model rather than a quoted energy error — "For all the three molecules, the deviation of the experimental results from the exact curves, is well explained by the stochastic simulations." No experimental energy is reported as reaching chemical accuracy: every chemical-accuracy figure in the paper is a simulation result. Noiseless simulation puts the critical depth for chemical accuracy (approx. 0.0016 Hartree) at d = 1, 8, 28 on the experimental connectivity and d = 1, 6, 16 all-to-all, for H2, LiH and BeH2, whereas on the hardware "the combined effect of decoherence and finite sampling sets the optimal depth for optimizations on our quantum hardware to 0 - 2 entanglers." Depth buys nothing measurable here: once decoherence, finite sampling and a limited number of trial states are combined, "the advantages of using more entanglers may not be apparent anymore", and for LiH at bond distance the single-run experiments at d = 0, 1 and 2 in Fig. S9a confirm the heavy overlap that the 100-outcome simulated histograms of Fig. S9b show between those depths — an overlap the paper says "appear[s] for most of the molecular Hamiltonians." At the critical depths of Fig. S7a, and by extrapolation rather than by running the larger simulations — an average energy standard deviation A computed on 10^2 random states at S = 10^3 samples, then scaled as A -> A*sqrt(10^3/S) — chemical accuracy is reached for all three molecules at S approx. 10^6. In a separate depolarizing-noise model of the 4-qubit LiH Hamiltonian at its bond distance, run to a ceiling of 5 x 10^4 function calls with each point averaged over 10 optimizations, chemical accuracy is reached only at noise rates of about 10^-5, and among the depths tried (0, 1, 2, 3, 4, 6 and 8 entanglers) only at 6 and 8.
Four-qubit Heisenberg magnet on the same device
The same processor and the same hardware-efficient trial states applied to a spin problem instead of a molecule: a four-qubit Heisenberg model on a square lattice in an external magnetic field, H = J sum_<ij> (X_i X_j + Y_i Y_j + Z_i Z_j) + B sum_i Z_i, where <ij> runs over nearest-neighbour pairs. The paper runs it to show "that with the same noisy quantum hardware, the advantage of using higher circuits depths is crucially dependent on the target Hamiltonian."
Four qubits, the same trial-state construction and SPSA feedback loop as the molecular runs, but with the circuit depth varied rather than held at d = 1: Fig. 4 compares d = 0 and d = 2 state preparations across a range of J/B, and Fig. S9c shows the experimental optimization at J/B = 1 for d = 0, 1, 2 and 3, one run per depth. Experimental results are compared against exact ground state energies and against density plots of 100 numerical outcomes that account for noisy energy estimation and decoherence.
The four-qubit Heisenberg Hamiltonian over a range of J/B ratios, with the detailed optimizations of Fig. 4a and Fig. S9c taken at J/B = 1.
At J = 0 the ground state is completely separable and the best estimates come from depth d = 0; as J is increased the ground state is increasingly entangled and the best estimates are instead obtained at d = 2, "despite the increased decoherence caused by using two entanglers for trial state preparation." Energies and, separately, the magnetization Mz are measured over the J/B range and compared with exact curves and the 100-outcome density plots. The paper's point is the contrast against its own molecular runs, where the depth advantage washes out: "In contrast, for the interacting spin Hamiltonians discussed in Fig. 4 of the main text, significantly better estimates are obtained with d = 1, 2, 3 circuits than d = 0 circuits."
What it needs
Nobody has taken this apart yet. That is a gap in this graph, not a claim that the method has no parts.
Other ways to fill the same slot
Different approaches
- Unitary coupled-cluster singles and doubles
Build the trial state from single and double excitations out of a reference determinant, exponentiated as a unitary. The family is chosen for chemistry rather than for the machine: the state you want is in it by construction, and the circuit that reaches it is deep.
- 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.
- k-UpCCGSD ansatz
Take k repetitions of paired double excitations together with generalized singles, instead of the full set of doubles. The point of the restriction is that the depth then grows linearly in the number of orbitals rather than polynomially, and k is the dial that buys accuracy back.
- 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.
- Particle-hole coupled-cluster circuits
Rewrite the Hamiltonian around the reference determinant so that what the circuit has to describe is excitations out of it, then build the family from gates that move an electron without creating or destroying one. Staying inside the right particle-number sector is a property of the gates, not something the optimiser has to discover.
- Orbital-optimized coupled-cluster circuits
Let the orbitals move too. The usual family fixes a basis and varies the amplitudes; this one varies the molecular orbital coefficients alongside them, so the same accuracy is reachable from a smaller active space and a shallower circuit — and the energy becomes fully variational, which is what makes forces available.
- Symmetry-preserving state-preparation circuits
Build the circuit so that it cannot leave the symmetry sector the chemistry lives in. Particle number, total spin, spin projection and time reversal are respected by the gate structure itself, so the search never spends parameters on states the answer cannot be in.
- 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.
- Generalized singles and doubles ansatz
Drop the rule that an excitation has to move an electron from an occupied orbital into an empty one. Every pair of orbitals may be coupled, so the circuit stops depending on which reference determinant it was built around — a wider variational manifold, paid for in parameters.
- 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
- Hardware-efficient VQE ansatz
Alternating native one-qubit rotations and entanglers reduce compilation overhead but can change trainability.