Qubit coupled-cluster method: A systematic approach to quantum chemistry on a quantum computer

Ilya G. Ryabinkin; Tzu-Ching Yen; Scott N. Genin; Artur F. Izmaylov

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Qubit coupled-cluster method: A systematic approach to quantum chemistry on a quantum computer

Ilya G. Ryabinkin, Tzu-Ching Yen, Scott N. Genin, Artur F. Izmaylov

arXiv:1809.03827v1Sep 11, 2018quant-phphysics.chem-ph

TL;DR

Quantum chemistry VQE methods need accurate correlation models that fit limited quantum coherence and entangling capabilities. This paper introduces QCC directly in qubit space, ranks entanglers by estimated energy contributions, and factorizes multi-qubit rotations into two-qubit rotations. QCC achieves chemical accuracy for H2 and LiH ground-state potential energy curves while using compact entangler sets.

Problem
UCC accuracy depends on selected excitations, while its transformed operators can require entangling more qubits than available architectures support.
Method
QCC builds the ansatz from qubit operators, screens entanglers by estimated correlation-energy contributions, and exactly factorizes multi-qubit rotations into two-qubit rotations.
Results
QCC achieved chemical accuracy for H2 and LiH ground-state potential energy curves using compact qubit-space parametrizations.
Takeaways & Limitations
The factorization enables QCC simulations on hardware limited to two-qubit entanglement, while the LiH calculation used only 15 variational parameters.

Abstract

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A unitary coupled-cluster (UCC) form for the wavefunction in the variational quantum eigensolver has been suggested as a systematic way to go beyond the mean-field approximation and include electron correlation in solving quantum chemistry problems on a quantum computer. Although being exact in the limit of including all possible coupled-cluster excitations, practically, the accuracy of this approach depends on how many and what kind of terms are included in the wavefunction parametrization. Another difficulty of UCC is a growth of the number of simultaneously entangled qubits even at the fixed fermionic excitation rank. Not all quantum computing architectures can cope with this growth. To address both problems we introduce a qubit coupled-cluster (QCC) method that starts directly in the qubit space and uses energy response estimates for ranking the importance of individual entanglers for the variational energy minimization. Also, we provide an exact factorization of a unitary rotation of more than two qubits to a product of two-qubit unitary rotations. Thus, the QCC method with the factorization technique can be limited to only two-qubit entanglement gates and allows for very efficient use of quantum resources in terms of the number of coupled-cluster operators. The method performance is illustrated by calculating ground-state potential energy curves of H$_2$ and LiH molecules with chemical accuracy, $\le 1$ kcal/mol.

I. INTRODUCTION

Quantum chemistry on quantum computers is motivated by the classical exponential cost of exact electronic-structure calculations and constrained quantum resources. VQE and UCC offer resource-aware alternatives, but UCC’s qubit locality and gate requirements motivate a directly qubit-based approach with systematic entangler selection and factorization.

Motivation: Exact electronic-structure calculations scale exponentially with system size classically, while quantum approaches seek polynomial scaling without introducing approximations.Quantum hardware remains constrained by limited numbers of fully interacting qubits and finite coherence times.
Existing approaches: QPE requires long coherent evolution for accurate phase estimation, making it difficult on currently available quantum computers.VQE instead variationally optimizes a unitary acting on an initial qubit state to estimate the ground-state energy.
UCC limitations: UCC borrows fermionic coupled-cluster parametrization for VQE, truncating excitation rank to obtain approximate treatments whose low-rank accuracy depends on electron correlation.Fermion-to-qubit transformations turn excitation operators into long Pauli-word combinations.
UCC limitations: UCC’s multi-qubit operations require increasing entanglement or factorization into one- and two-qubit gates, but no general efficient factorization had been proposed.This creates an architectural challenge because not all quantum computers can directly implement operations involving many qubits.
Paper contribution: Hardware-efficient ansätze use only two-qubit entanglers, but fixed generators may converge slowly because the system, rather than hardware, determines efficient entanglers.The paper therefore introduces QCC, ranks entanglers by estimated correlation-energy contributions, and derives exact multi-qubit factorization.

A. Qubit Coupled-Cluster method

QCC constructs the variational wavefunction directly from qubit operators: single-qubit coherent states provide the mean-field component, while Pauli-word rotations introduce correlation. Entangler screening supports systematic selection, and exact factorization enables implementation with two-qubit gates, although the unselected fully transformed Hamiltonian can still have exponential operator growth.

Qubit Coupled-Cluster method: QCC uses a product of single-qubit coherent states for the mean-field wavefunction and multi-qubit Pauli-word rotations for correlation.The coherent states are parametrized by Bloch-sphere angles, while entanglers have real amplitudes.
Qubit Coupled-Cluster method: Unlike UCC generators, which become lengthy Pauli-word combinations after fermion-to-qubit transformation, QCC generators are individual involutory Pauli words.Their simple algebra yields compact expressions for similarity-transformed operators.
Complexity: Without restricting the operator treatment, the fully transformed Hamiltonian can involve approximately 3^Nent operators, revealing exponential complexity.This motivates selecting a limited entangler set rather than retaining every possible transformed term.
Qubit Coupled-Cluster method: QCC requires 2Nq + Nent variational parameters: two Bloch angles per qubit plus one amplitude for each optimized entangler.This gives a compact parameterization in the qubit space.

B. Entanglers’ ranking

QCC ranks entanglers by their estimated energy lowering, using derivative-based prescreening to reduce the cost of evaluating candidates. The final selection distinguishes first- and second-derivative signals before comparing top candidates by full energy lowering.

The ranking quantity is the energy lowering ΔE[P_k] obtained by optimizing each candidate entangler against the qubit mean-field reference.The qubit mean-field energy E_QMF equals the zero-amplitude energy E[0; P_k].
Prescreening uses the first and second terms of each similarity-transformed energy’s Taylor expansion.The mean-field Bloch angles relax with the entangler amplitude, so they must be treated as implicit functions of τ.
The mean-field state is parameterized by Bloch angles {θ1, φ1, ..., θNq, φNq}, optimized before evaluating entangler responses.
Important entanglers form two tiers: nonzero first derivatives, or zero first derivatives with significant negative second derivatives.Final ranking evaluates ΔE[P_k] for top entanglers from both tiers.

C. Factorization of multi-qubit entanglers

The factorization procedure converts multi-qubit entanglers into products of shorter-generator unitary transformations, addressing hardware limits on directly entangling more than two qubits. The resulting number of two-qubit factors grows logarithmically with the original Pauli-word length without increasing variational parameters.

Hardware limitations can prevent direct implementation of important three- or four-body entanglers identified by the ranking procedure.
The recursive factorization replaces a many-qubit unitary with three unitaries whose Pauli-word generators involve fewer qubits.
The elementary step starts from a Pauli word of length |P| ≥ 3 and partitions it into non-overlapping components around a selected single-qubit operator.The components commute because they act on non-overlapping qubit sets.
The number of two-qubit factors grows as approximately log2|P|, while the number of variational parameters remains unchanged.Factorization increases the number of entanglers, not variational parameters.

III. NUMERICAL STUDIES AND DISCUSSION

The numerical study evaluates QMF and QCC ground-state potential energy curves for H2 and LiH. LiH provides a case where electronic correlation entangles electrons across more than two orbitals.

Ground-state potential energy curves were computed for H2 and LiH using both QMF and QCC approaches.The molecules had previously been used to illustrate quantum-computing techniques.
LiH was selected because it is among the simplest molecules whose electronic correlation entangles electrons on more than two orbitals.

A. Fermionic Hamiltonian quantities

The study generated fermionic Hamiltonian quantities with a modified GAMESS package and used RHF orbitals in the STO-3G basis with molecule-specific symmetries. H2 included all four spin-orbitals, while LiH used an active-spin-orbital subset.

A locally modified GAMESS package generated fermionic Hamiltonian spin-orbitals and one- and two-electron integrals.
Calculations used canonical RHF orbitals in the STO-3G basis, assuming D2h symmetry for H2 and C2v symmetry for LiH.
All four H2 spin-orbitals entered the second-quantized Hamiltonian, whereas LiH was represented using a selected active subset.

B. Generation of qubit operators

The molecular Hamiltonians are mapped into qubit space using different fermion-to-qubit transformations, with stationary qubits enabling further reduction for LiH.

H2 molecule: The H2 Hamiltonian uses the Bravyi–Kitaev transformation and retains four spin-orbitals without exploiting two stationary qubits.The spin-orbitals are ordered alternately as α, β, α, … .
LiH molecule: The LiH Hamiltonian uses the parity transformation with spin-orbitals ordered as all alpha followed by all beta.Three active molecular orbitals produce a six-qubit Hamiltonian containing 118 Pauli words at each bond distance.
LiH molecule: LiH’s third and sixth stationary qubits can be replaced by eigenvalues ±1, producing sectors with four-qubit effective Hamiltonians.The ground state lies in the z2 = −1, z5 = 1 sector, whose effective Hamiltonian has 100 Pauli terms.
Entangler ranking: The entangler-ranking table evaluates two-qubit operators for H2 at R = 1.0 Å using first- and second-order energy responses.The table reports quantities in Hartree atomic units.

C. Potential energy curves

QCC potential-energy calculations achieve chemical accuracy using screened entanglers, while symmetry restoration is important for LiH over the full bond-length range. The factorization procedure converts larger entanglers into two-qubit operations without increasing variational parameters.

Symmetry considerations: QMF solutions can break singlet spin symmetry, whereas imposing ⟨S2⟩ = 0 restores the correct symmetry and smooths the potential-energy surfaces.The QCC calculations began with symmetry-unconstrained QMF solutions.
H2 potential-energy curve: Only one ranked two-qubit entangler is needed for H2 in the minimal basis to reach the exact energy across the potential-energy curve.Among 54 two-qubit entanglers, six lower the energy, but only two have nonzero first-order gradients.
H2 potential-energy curve: The H2 QCC circuit uses single-qubit rotations, Hadamard gates, and an entangler implemented with elementary Rigetti operations.The entangler amplitude controls the interleaved RZ gate, while Bloch angles determine the spin-coherent-state rotations.
LiH potential-energy curve: Six to seven entanglers provide chemical accuracy for LiH near R(Li−H) ≈ 1.5 Å, but they do not suffice across 0.5–5.0 Å.The region R(Li−H) ≳ 3.2 Å is especially demanding because QMF experiences symmetry breaking.
LiH potential-energy curve: Using seven entanglers selected for energy lowering and spin-symmetry restoration reaches chemical accuracy across the LiH potential-energy curve.These symmetry-affecting entanglers can retain QMF symmetry where restoration is unnecessary through variational amplitude optimization.
Factorization and resources: Factorizing three- and four-qubit LiH entanglers produces 31 two-qubit entanglers without changing the number of variational parameters.The resulting LiH QCC ansatz has 15 variational parameters.

IV. CONCLUSIONS

QCC formulates coupled-cluster optimization directly with qubit operators, using compact Pauli-word transformations, two Bloch angles per qubit, and a two-qubit factorization. It also addresses qubit-space complications through constraints and a derivative test, achieving chemical accuracy for H2 and LiH with a small parametrization.

QCC is a variational method formulated exclusively with qubit operators rather than fermionic algebra.
Two Bloch angles per qubit parametrize single-qubit rotations, while Pauli-word transformations provide compact matrix-element expressions.
Multi-qubit Pauli-word rotations can be exactly decomposed into products of two-qubit unitary rotations.
QCC addresses qubit-space complications by using constraints for physical symmetries and a derivative test to tailor the ansatz to each molecular system.
Seven entanglement generators and 15 variational parameters achieved chemical accuracy across the H2 and LiH ground-state dissociation curves.The parametrization comprised 8 Bloch angles and 7 generator amplitudes.

Appendix: Qubit coupled-cluster energy derivatives

The appendix derives energy derivatives for optimizing QCC entangler amplitudes while allowing Bloch angles to vary with those amplitudes. It also describes practical procedures for undefined angles and discontinuous derivatives.

The first derivative of the similarity-transformed energy follows from cancellation of terms enforced by Bloch-angle optimality.
The QCC energy is expanded around the mean-field solution in entangler amplitudes and Bloch-angle displacements.The expansion uses derivatives with respect to entangler amplitudes and Bloch angles.
Optimized Bloch angles become functions of the entangler amplitude, with their dependence determined by linear equations from the minimization condition.
Higher-order coherent-state derivatives can be generated successively because single-qubit coherent states form a complete two-dimensional basis.
Undefined azimuthal angles at polar angles 0 or π are determined by re-optimizing angles at small entangler amplitudes.The appendix gives τ = 10^-3 as an example small amplitude.
When first derivatives differ significantly around τ = 0, the second derivative is not evaluated because a derivative discontinuity is indicated.The discontinuity is tested using small positive and negative amplitudes.