Unitary coupled-cluster singles and doubles

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

O'Malley et al. report this as the first electronic structure calculation on a quantum computer "without exponentially costly precompilation", running UCC through VQE alongside the Trotterisation-and-phase-estimation route on the same device, and comparing them: they find "clear evidence that the variational quantum eigensolver is robust to certain errors". That comparison is the evidence for the whole variational branch, and it was made on molecular hydrogen.

Requires

Every step this method names moves its route along, so there is nothing it needs alongside them.

Example

given  ground-state eigenvalue problem for Hamiltonian H on N spin orbitals, eta electrons
       Hartree-Fock reference determinant  |phi>

requires  a reference state |phi> -- here taken to be the Hartree-Fock
    determinant; that is a choice specific to this construction, not a
    general requirement of UCC, which the paper notes (unlike traditional
    CC) can instead be built on a multireference reference          (Appendix D)
    # for this HF choice, occ / virt below is the fixed split of spin
    #   orbitals that determinant defines

# --- cluster operator, truncated to singles and doubles ---------------------
T(theta) = T1(theta) + T2(theta)
    # the full cluster operator is T = sum_k T^(k)(theta); stopping at k=1,2
    #   is what earns this ansatz the name UCCSD, as opposed to full UCC   (Appendix D)
T1(theta) = sum_{i in occ, a in virt}  theta^a_i  a_a^dagger a_i                (Eq. D3)
T2(theta) = (1/4) sum_{i1,i2 in occ, a1,a2 in virt}
                theta^{a1,a2}_{i1,i2}  a_a2^dagger a_i2  a_a1^dagger a_i1        (Eq. D4)

# --- exponentiate as a unitary onto the reference ----------------------------
U(theta) = exp( T(theta) - T(theta)^dagger )
|phi(theta)> = U(theta) |phi>                                                    (Eq. D1)
    # anti-Hermitian generator => U(theta) unitary for EVERY theta, so the
    #   variational principle holds at any theta, not only at convergence:
    #   E(theta) = <phi(theta)|H|phi(theta)> >= E_0                     (Eq. 2)
    # classical CC's non-unitary e^T has no such guarantee -- with spin-
    #   restricted orbitals, CCSD's projective energy is "not in general
    #   variational" and catastrophic failure on bond-breaking is quite
    #   common                                                    (Lee et al.)
    # for H2 in a minimal basis exactly one amplitude survives: after the
    #   qubit mapping this specializes to  |phi(theta)> = exp(-i theta X0 Y1) |01>
    #                                                                       (Eq. 3)

# --- compile onto a device: Trotterize term by term --------------------------
U_trot(theta) = prod over each excitation/de-excitation term tau of T(theta)-T(theta)^dagger
                    of  exp( theta_tau * tau )
|phi(theta)> ~= U_trot(theta) |phi>          # this, not U(theta), is what runs on hardware
    # a practical circuit uses a SMALL number of Trotter steps and is therefore
    #   not exactly equivalent to the ideal e^{T-T^dagger} ansatz above; one step
    #   has been reported -- not proven in general -- to already reach chemical
    #   accuracy                                              (Lee et al., citing ref. 64)
    # Lee et al. count O((N-eta)^2 eta^2) such terms, each acting on O(1) spin
    #   orbitals, and note gate count would naively be expected to be lower-
    #   bounded by the free amplitudes theta^a_i, theta^{a1,a2}_{i1,i2} --
    #   possibly with added overhead from the fermion-to-qubit mapping and
    #   device connectivity                                       (Lee et al., p.16)
    # no analogous EXACT preparation is known to be efficient classically; the
    #   scalable routes cited here are for a fixed, already-chosen theta on
    #   quantum hardware                                             (Appendix D)

return  the parameterized circuit family  |phi(theta)> ~= U_trot(theta) |phi>
# theta is left free here -- a separate, outer classical loop (VQE) varies it
#   to minimize E(theta); that loop is not part of building this family

Cost, as the source states it

Circuit depth $\mathcal{O}((N-\eta)^2\eta)$ in the number $N$ of spin orbitals and the number $\eta$ of electrons — the figure Lee et al. quote for UCCSD when placing their own ansatz against it. It is a depth for the family, not a complexity for solving the problem: no worst-case speedup is claimed here or anywhere in this region.

Implementations

• UCC-VQE hydrogen curve on the Xmon superconducting processor

  • Scalable Quantum Simulation of Molecular Energies

  O'Malley et al. set out to compute the potential energy curve of molecular hydrogen on superconducting hardware and to reach chemical accuracy, which they define as $1.6\times10^{-3}$ Hartree — the relative energy error that at room temperature already shifts a chemical rate by an order of magnitude. They report it as "the first electronic structure calculation performed on a quantum computer without exponentially costly precompilation", the point being that earlier chemistry experiments represented the Hamiltonian in a configuration basis and exponentiated an exponentially large matrix as a classical preprocessing step. The same device ran the Trotterisation-plus-phase-estimation route on the same molecule, so the two approaches could be compared directly; only the UCC-VQE half is this implementation.

  Two qubits of a linear Xmon chain carry the whole molecule. The Hartree-Fock state $|01\rangle$ is prepared with an $X_\pi$ gate on $q_0$, and the unitary coupled cluster ansatz is applied as $|\varphi(\theta)\rangle = e^{-i\theta X_0 Y_1}|01\rangle$ (Eq. 3); the paper states outright that "for molecular hydrogen in the minimal basis, there is exactly one term in the UCCSD ansatz" (Appendix D), so the circuit carries a single variational parameter. The rotation $Z_\theta$ is implemented as a phase shift on all subsequent gates; a single VQE sequence is 11 single-qubit gates and two $CZ_\pi$ gates. Rather than close the classical feedback loop, the authors "elected to scan a thousand different values of $\theta \in [-\pi,\pi)$" at each bond length, then took the smallest energy; they state this was done to simplify the classical feedback routine at the cost of more experimental trials, and that the energy surface of Fig. 2b "was locally optimized at each bond length to emulate an on-the-fly implementation". Energies are assembled from partial tomography of the terms of the two-qubit Hamiltonian — $X_1X_0$, for instance, is measured by applying $Y_{\pi/2}$ gates to each qubit before readout — and summed with the tabulated coefficients (Eq. 4). The experiment is run in different gauges of the Bravyi-Kitaev transform (a flip of one or both qubits in the Hartree-Fock input state plus a sign change on the relevant Hamiltonian terms) and the statistics averaged together.

  Molecular hydrogen in the minimal STO-6G basis of Hartree-Fock orbitals, second-quantised and mapped to qubits by the Bravyi-Kitaev transformation; the four-qubit spin Hamiltonian is quoted as reported by Seeley, Richard and Love (J. Chem. Phys. 137, 224109 (2012)). Qubits 1 and 3 are never flipped from the Hartree-Fock start, and that symmetry reduces the problem to the two-qubit effective Hamiltonian $\tilde{H} = g_0\mathbb{1} + g_1 Z_0 + g_2 Z_1 + g_3 Z_0Z_1 + g_4 X_0X_1 + g_5 Y_0Y_1$ (Eq. A6). Table I gives the six coefficients at every bond length from $R = 0.20$ to $2.85$ Å in steps of 0.05 Å, together with the Trotter orderings and $t_0$ used by the phase-estimation half of the paper. The authors note in the acknowledgements that an error in the $g_0$ values of an earlier version of the work was brought to their attention and corrected.

  Run on Xmon variant planar transmon qubits in a dilution refrigerator at a base temperature of 20 mK; for VQE, $q_0$ and $q_1$ at 4.49 and 5.53 GHz with all other qubits detuned to 3 GHz and below, energy relaxation times $T_1 = 62.8$ and $21.4\,\mu$s and Ramsey decay times $T_2^* = 1.1$ and $1.9\,\mu$s. The VQE experiment predicts the dissociation energy with an error of $(8 \pm 5)\times10^{-4}$ Hartree, below the chemical accuracy threshold. That dissociation energy is taken relative to the equilibrium geometry, which the paper identifies with the minimum-energy bond length $R = 0.72$ Å of the curve in Fig. 3a — not a bond length the experiment sampled, since Table I's grid runs from 0.20 to 2.85 Å in steps of 0.05 Å. The same hardware run evaluated at the theoretically optimal $\theta$ instead of the experimentally optimal one gives a dissociation energy error of $1.1\times10^{-2}$ Hartree, "more than an order of magnitude worse" — the paper calls this "the first experimental signature of robustness" of VQE to systematic error, and the comparison with its own PEA run, which showed a dissociation energy error of $(1 \pm 1)\times10^{-2}$ Hartree from a single Trotter step, is what the paper leans on for "clear evidence that the variational quantum eigensolver is robust to certain errors". Error bars come from a Gaussian process regression over the potential energy curve, propagated from shot-noise-limited expectation values.

• In-house UCCSD benchmark code over H4, H2O and N2

  • Generalized Unitary Coupled Cluster Wavefunctions for Quantum Computation

  Lee et al. introduce one new ansatz, $k$-UpCCGSD, and need two points of comparison for it: UCCGSD — a unitary generalized-singles-and-doubles ansatz that they name and are the first to study classically without approximating the energy evaluation, but whose idea they credit to Nooijen — and UCCSD as the incumbent. The comparison runs on resources and on accuracy. Their resource table is where this method's depth figure comes from: for a fixed number of Trotter steps, UCCSD costs $\mathcal{O}((N-\eta)^2\eta^2)$ gates at circuit depth $\mathcal{O}((N-\eta)^2\eta)$, against $\mathcal{O}(N^4)$ and $\mathcal{O}(N^3)$ for UCCGSD and $\mathcal{O}(kN^2)$ and $\mathcal{O}(kN)$ for $k$-UpCCGSD (Table 1). The accuracy half is what this entry records: three molecules of very different symmetries, each run through UCCSD and through the other two ansätze on the same footing.

  The energy is computed variationally as a function of the cluster amplitudes, $E(\{t^a_i\},\{t^{ab}_{ij}\}) \equiv \langle\psi|\hat{H}|\psi\rangle / \langle\psi|\psi\rangle$ (Eq. 9), and its gradient handed to SciPy's implementation of the BFGS algorithm, a quasi-Newton method that needs no explicit Hessian. All the UCCSD calculations run in an in-house code the authors describe as using OpenFermion together with TensorFlow for efficient gradient evaluations; its stated ceiling is "about 16 spin-orbitals", which the authors say allowed them to examine the model systems presented, and they point readers wanting a production-level code to the implementation of Evangelista (J. Chem. Phys. 134, 224102 (2011)). Because it is not clear whether a given UCC class reaches its global minimum, "each gradient-based optimization was therefore carried out between thirty and two hundred times (depending on the cost) starting from randomly chosen initial points". The frozen core approximation is applied to oxygen and nitrogen. Accuracy is reported as the non-parallelity error, defined here as the difference between the maximum and the minimum error over the curve. Excited states are obtained with a variant of the orthogonally constrained VQE (OC-VQE) framework of Higgott, Wang and Brierley (arXiv:1805.08138). The authors are explicit that this is a classical benchmark of the ideal ansatz, not of a compiled circuit: "a practical implementation of UCC relies on approximating $e^{\hat{T}-\hat{T}^\dagger}$ by a small number of Trotter steps, which leads to ansätze that are not exactly equivalent to the ones considered in our numerical calculations", and BFGS as performed here "is unsuitable for use on a quantum device due to the stochastic error associated with the measurement of observables in the VQE framework".

  Three systems. H4 in a coordinate setup with fixed $R_{\mathrm{H-H}} = 1.23$ Å and the separation $R$ between the two H2 units varied, in STO-3G ($N = 8$, $\eta = 4$) and in 6-31G ($N = 16$, $\eta = 4$) — the 6-31G case is called "the most expensive calculation reported in this work" in quantum-device terms. The double dissociation of H2O at fixed $\theta_{\mathrm{HOH}} = 104.5^\circ$ within $C_{2v}$ symmetry, STO-3G ($N = 12$, $\eta = 8$). The dissociation of N2 in STO-3G ($N = 16$, $\eta = 10$). Full configuration interaction reference energies come from Psi4 through its OpenFermion interface; the classical coupled-cluster comparisons (RCCSD, UnrCCSD, CCVB-SD) come from the development version of Q-Chem.

  All UCCSD calculations were performed with the authors' in-house code, which uses OpenFermion together with TensorFlow for efficient gradient evaluations; everything else the demonstrations required was done with the development version of Q-Chem. Both are classical computations — no quantum hardware and no quantum simulator is involved, and the paper reports no machine, cluster or wall-clock figure. All figures below are non-parallelity errors in mE$_h$. On the ground state of H4 in STO-3G, "except 1-UpCCGSD and UCCSD, all the UCC variants are numerically exact". On the first excited state of H2O in STO-3G, UCCSD's NPE is 17.57, against 0.00 for UCCGSD, 30.22 for 1-UpCCGSD, 0.98 for 2-UpCCGSD and 0.01 for 3-UpCCGSD (Table 5b) — 1-UpCCGSD is worse than UCCSD, and the paper groups the two together as the methods that "do not deliver reliable excited state energies". On the first excited state of N2 in STO-3G, UCCSD's NPE is 31.94, worse than UCCGSD's 7.79, and neither is near chemical accuracy; on the ground state of N2, UCCSD is worse in NPE than unrestricted CCSD, whose NPE is 8.98. For H4 in 6-31G the authors find RCCSD better than UCCSD and suspect that is fortuitous, since RCCSD "clearly becomes non-variational at $R = 1.23$ Å". The paper's own reading of the sweep is that "UCCSD does not perform very well given the number of amplitudes" and that $k$-UpCCGSD at a comparable amplitude count always beat it.

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

• 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.

• 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

• UCCSD VQE ansatz

  A chemistry-inspired unitary coupled-cluster ansatz truncated to single and double excitations.

Sources

• Scalable Quantum Simulation of Molecular Energies↗P. J. J. O'Malley, R. Babbush, I. D. Kivlichan, J. Romero, J. R. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. Ding, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Jeffrey, A. Megrant, J. Y. Mutus, C. Neill, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, P. V. Coveney, P. J. Love, H. Neven, A. Aspuru-Guzik, J. M. Martinis · 2015theory no · simulation ? · hardware yes
• Generalized Unitary Coupled Cluster Wavefunctions for Quantum Computation↗Joonho Lee, William J. Huggins, Martin Head-Gordon, K. Birgitta Whaley · 2018theory yes · simulation yes · hardware no