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Quantum algorithms for electronic structure calculations: particle/hole Hamiltonian and optimized wavefunction expansions

Panagiotis Kl. Barkoutsos, Jerome F. Gonthier, Igor Sokolov, Nikolaj Moll, Gian Salis, Andreas Fuhrer, Marc Ganzhorn, Daniel J. Egger, Matthias Troyer, Antonio Mezzacapo, Stefan Filipp, Ivano Tavernelli

arXiv:1805.04340v1quant-ph

TL;DR

The paper addresses efficiency and scalability limits in quantum algorithms for molecular ground-state calculations. It reformulates the Hamiltonian in the particle-hole picture and develops UCC and heuristic wavefunction Ansätze with exchange-type circuits. Simulations report that VQE can reproduce ground-state energies with a single Trotter step, including errors below 10^-10 Ha.

  • Problem

    Quantum chemistry methods and classically inspired trial states face unfavorable scaling or rapidly increasing parameter counts, motivating more efficient quantum formulations.

  • Method

    The paper combines a particle-hole Hamiltonian with coupled-cluster and heuristic VQE Ansätze, including particle-conserving exchange-type entangler blocks and active-space reductions.

  • Results

    The VQE implementation of UCC can use a single Trotter step while retaining ground-state-energy errors below 10^-10 Ha in simulations.

  • Takeaways & Limitations

    The particle-hole representation improves the starting point for wavefunction optimization, while q-UCC provides a variational quantum version of the classical coupled-cluster approach.

Abstract

from arXiv · show

In this work we investigate methods to improve the efficiency and scalability of quantum algorithms for quantum chemistry applications. We propose a transformation of the electronic structure Hamiltonian in the second quantization framework into the particle-hole (p/h) picture, which offers a better starting point for the expansion of the trial wavefunction. The state of the molecular system at study is parametrized in a way to efficiently explore the sector of the molecular Fock space that contains the desired solution. To this end, we explore several trial wavefunctions to identify the most efficient parameterization of the molecular ground state. Taking advantage of known post-Hartree Fock quantum chemistry approaches and heuristic Hilbert space search quantum algorithms, we propose a new family of quantum circuits based on exchange-type gates that enable accurate calculations while keeping the gate count (i.e., the circuit depth) low. The particle-hole implementation of the Unitary Coupled Cluster (UCC) method within the Variational Quantum Eigensolver approach gives rise to an efficient quantum algorithm, named q-UCC , with important advantages compared to the straightforward 'translation' of the classical Coupled Cluster counterpart. In particular, we show how a single Trotter step can accurately and efficiently reproduce the ground state energies of simple molecular systems.

I. INTRODUCTION

The paper frames quantum chemistry as a search for scalable quantum algorithms, focusing on particle-hole reformulation and efficient trial-state parameterizations. The particle-hole picture uses the Hartree-Fock state as a reference vacuum for expansions around molecular ground states.

  • Classical electronic-structure methods can reach arbitrary precision but scale unfavorably, from O(N!) for full CI to O(N^7) for CCSD(T).
  • Quantum chemistry algorithms must map the electronic Hamiltonian to qubit operators, prepare suitable trial wavefunctions, and optimize their parameters.
  • Second quantization encodes electronic degrees of freedom in wavefunction expansion coefficients, avoiding first-quantization spatial discretization costs.
  • The work investigates particle-hole Hamiltonians, trial-state parameterizations, and particle-conserving exchange-type gates to improve quantum-algorithm efficiency.
  • The particle-hole transformation treats the Hartree-Fock determinant as a vacuum and describes excitations through hole and particle quasiparticles.
  • Redefining normal ordering in the particle-hole picture enables an efficient perturbative expansion using Wick’s theorem, independent of electron number.

B. Trial wavefunctions

The paper distinguishes classically inspired coupled-cluster trial states from heuristic states that directly sample relevant regions of Hilbert space. These approaches differ in how they parameterize and search molecular wavefunctions.

  • Trial wavefunctions apply parameterized excitations to the Hartree-Fock state, with gate angles optimized until convergence.
  • The coupled-cluster Ansatz offers a controllable expansion with an efficient parameterization that minimizes independent parameters.
  • The unitary coupled-cluster formulation is better suited to quantum computing because of the properties of its gate operations.
  • Heuristic trial states lack a strict classical equivalent and aim to sample efficiently the Hilbert-space region containing the solution.

1. The UCC Ansatz

The UCC approach maps truncated unitary coupled-cluster excitations into parameterized quantum circuits and studies approximations that reduce their depth. Within VQE, the implementation is restricted to singles and doubles and examines Trotterization.

  • The UCC trial state uses an excitation operator expanded as T(θ)=T1(θ)+T2(θ)+⋯+Tn(θ), with θ collecting expansion coefficients.
  • The implementation restricts UCC to singles and doubles, while higher excitations require longer circuits considered impractical for experiments and simulations.
  • VQE optimizes θ using the particle-hole Hamiltonian to obtain the correlated ground-state energy relative to the Hartree-Fock energy.
  • Exponentiating UCCSD operators into circuits is challenging because the resulting circuit depth scales with occupied and virtual orbital counts.
  • The variational UCCSD formulation differs from classical CCSD because replacing Ti with (Ti−Ti†) removes the classical closed-equation treatment.
  • Single and double excitation operators are implemented by the circuits shown in Fig. 1, contributing respectively to T1 and T2.

2. The Heuristic Ansatz

The heuristic Ansatz combines single-qubit rotations with entangling blocks, emphasizing particle-conserving exchange gates and the particle-hole Hamiltonian. The design controls Hilbert-space exploration through gate structure and an optional particle-number penalty.

  • The heuristic VQE state alternates single-qubit rotations and entangling operations applied to the Hartree-Fock reference.
  • The method uses particle-conserving entanglers to constrain the search to a fixed particle-number sector and exploit exchange-type hardware gates.
  • Three entangler blocks are investigated: exchange-type gates with two parameters, a single-parameter exchange gate, and an all-to-all CNOT block.
  • The first two exchange-type gates can implement a particle-conserving SWAP directly in hardware in one step.
  • The all-to-all CNOT entangler does not conserve particle number, allowing optimization through Fock-space regions with different electron counts.
  • A tunable number-operator potential can constrain the final electron count and be introduced gradually to retain early optimization flexibility.

C. Reduction of the Hilbert space

The paper reduces quantum-chemistry simulation resources by replacing inert core electrons with effective core potentials and restricting UCC expansions to an active orbital space. The resulting workflow maps, prepares, measures, and classically optimizes particle/hole Hamiltonians until convergence.

  • Effective Core Potentials: Effective core potentials replace inert innermost-shell electrons, reducing the number of degrees of freedom and required Hartree–Fock orbitals.This consequently reduces the number of qubits used to expand the electronic and particle/hole Hamiltonians.
  • Selection of the Active Space: Active-space selection restricts UCC searches to subsets of occupied and virtual orbitals defined by Nocc and Nvir.The selected orbital ranges determine which reduced excitation operators enter the expansion.
  • Selection of the Active Space: Using an active space shortens the overall circuit depth, improving use of limited qubit coherence time.
  • VQE workflow: The VQE workflow computes Hartree–Fock orbitals, applies Jordan–Wigner mapping, prepares trial states, measures the particle/hole energy, and updates parameters with BFGS.Steps (iii) through (v) repeat until convergence.
  • VQE workflow: Trial states start from the Hartree–Fock ground state and use the UCCSD or heuristic circuits, with initial gate angles sampled uniformly from 0 to 2π.The quantum computer evaluates the particle/hole Hamiltonian expectation value for classical optimization.
  • VQE workflow: The BFGS optimizer returns updated gate-angle values, and the preparation, measurement, and optimization stages iterate until convergence.

IV. RESULTS AND DISCUSSION

The H2 and H2O simulations evaluate particle/hole and conventional Hamiltonians with UCCSD and heuristic entangler circuits. The particle/hole formulation improves optimization efficiency and reduces heuristic resources, although CNOT entanglers can require substantially more blocks.

  • Simulation setup: H2 and H2O simulations use the 6-31G basis set, with oxygen core electrons replaced by effective core potentials so H2O includes eight valence electrons.The corresponding Hilbert-space dimensions are 2^8 for H2 and 2^12 for H2O.
  • UCCSD: 53 to 27 BFGS iterations and 3383 × Ns to 1471 × Ns circuit measurements are required when UCCSD uses the particle/hole Hamiltonian instead of the plain Hamiltonian.Both comparisons target convergence of 10^-7 Ha.
  • UCCSD: A factor 2 to 3 overall speed-up is observed for the particle/hole implementation of UCCSD.The UCCSD circuits and gate counts are the same for the two Hamiltonians, so the gain comes from more efficient parameter optimization.
  • Heuristic wavefunctions: 6 entangler blocks achieve 10^-7 Ha accuracy with the particle/hole Hamiltonian, compared with 8 blocks for the plain Hamiltonian at equilibrium.
  • Heuristic wavefunctions: 112 to 84 parameters and 56 to 42 gate operations are reduced by the particle/hole formulation for the U(1)ent heuristic approach.
  • Heuristic wavefunctions: A factor 3 to 4 overall gain is observed for the heuristic particle/hole implementation when chemical accuracy is required across the full dissociation path.
  • Limitations: CNOT entanglers generally require more resources, and CNOT/Full cannot reach chemical accuracy with fewer than 18 blocks in some cases.

B. The UCCSD Ansatz

The UCCSD Ansatz uses active-space restrictions and Trotterized excitation operators to reduce circuit complexity, while VQE reoptimization can preserve highly accurate energies even with one Trotter step.

  • Active-space restriction: Active-space restrictions reduce the UCCSD Hilbert-space search while retaining a one-to-one correspondence with the classical UCCSD algorithm.The approach investigates active spaces ranging from reduced orbital subsets to the full space.
  • Active-space restriction: For H2, AS4 already produces a qualitatively correct dissociation curve, and increasing the active space progressively approaches chemical accuracy.Chemical accuracy is defined as an error of 0.5 × 10^-2 Ha.
  • Active-space restriction: For H2O, active-space calculations substantially correct the Hartree-Fock profile, with errors below chemical accuracy below 1 Å and above 2 Å.The largest deviations occur at intermediate bond lengths near the Coulson-Fisher point, where spin-symmetry breaking can occur.
  • Trotter approximation: Trotter factorization introduces an approximation whose expansion becomes exact as the number of Trotter steps n approaches infinity.The simulations separate errors from second-order UCCSD truncation and from finite-step Trotter decomposition.
  • Trotter approximation: After VQE reoptimizes parameters at each n, the Trotter error is negligibly small and remains below 10^-10 Ha even for n = 1.Using optimized angles without reoptimization gives the analytic Trotter dependence, whereas full VQE reoptimization yields the near-zero-error result.
  • Trotter approximation: The resulting q-UCCSD implementation absorbs Trotter error variationally, enabling a single-step Trotter expansion and reduced circuit depth.This variational implementation no longer maintains a strict one-to-one mapping to the classical CCSD expansion.

C. The heuristic Ansatz

The heuristic VQE Ansatz combines particle–hole Hamiltonians with specialized entangler blocks to compute molecular dissociation profiles. Particle-conserving entanglers perform similarly, while the non-particle-conserving circuit shows larger deviations and misses chemical accuracy for H2 at larger distances.

  • Heuristic Ansatz: The heuristic VQE approach combines the particle–hole Hamiltonian with three specialized entangler blocks, U^(1–3).The number of blocks is selected to reach 10^-7 Ha accuracy at equilibrium, with dissociation profiles evaluated for H2 and H2O.
  • Heuristic Ansatz: The non-particle-conserving entangler U^(3) shows larger deviations across the distance range and may suffer from sampling or convergence difficulties.Its variable-electron-number Fock-space search is identified as one possible source of the discrepancy.
  • Heuristic Ansatz: With the quoted number of entangler blocks, the heuristic and UCCSD calculations do not achieve chemical accuracy for H2 at large distances R > 1.3 Å.The limitation is reported for the dissociation regime, whereas the water-molecule results are described separately.

V. CONCLUSIONS

The conclusions report that particle–hole reformulation improves VQE optimization and that both q-UCC and heuristic Ansätze can reduce quantum-resource requirements. A single Trotter step gives highly accurate simulated energies, while exchange-type gates improve heuristic circuit efficiency.

  • V. CONCLUSIONS: The particle–hole Hamiltonian shifts the reference from the vacuum to the Hartree–Fock wavefunction, accelerating convergence of the correlation energy.Restricting to chemically active valence electrons with effective core potentials can also achieve chemical accuracy using fewer qubits.
  • V. CONCLUSIONS: A single Trotter step approximates the exponentiated cluster operators with simulated ground-state-energy errors below 10^-10 Ha.The variational flexibility of VQE can absorb the Trotter error, motivating the q-UCC name for this implementation.
  • V. CONCLUSIONS: Using one Trotter step reduces the number of gates required for the coupled-cluster implementation by about a factor 10^3.The reduction is identified as important for implementation on real quantum hardware.
  • V. CONCLUSIONS: Specialized exchange-type two-qubit gates substantially improve the efficiency of heuristic entangler blocks.The paper presents these gates as part of a heuristic wavefunction-expansion approach.

Appendix A: Approximations in UCCSD

The appendix derives approximations for the UCCSD expansion by restricting the excitation operator to singles and doubles, applying first-order Trotterization, and mapping fermionic operators to qubits.

  • Appendix A: Approximations in UCCSD: The UCCSD excitation operator is restricted to single and double excitations, with real parameters θ_ij and θ_ijkl.This restriction defines the operator used in the subsequent approximation.
  • Appendix A: Approximations in UCCSD: First-order Trotterization factorizes exponentials of sums into ordered products of exponentials for the individual excitation terms.The appendix applies the approximation repeatedly to construct the UCCSD circuit operator.
  • Appendix A: Approximations in UCCSD: The Jordan–Wigner transformation maps fermionic creation and annihilation operators to Pauli operators on N_q qubits.The resulting strings contain X, Y, and Z operators whose ordering depends on the orbital indices.
  • Appendix A: Approximations in UCCSD: For selected operator pairs whose nested commutators vanish, Glauber’s formula permits an exact exponential factorization rather than an approximation.The appendix uses this condition to simplify parts of the q-UCCSD expansion operator.
  • Appendix A: Approximations in UCCSD: Excitation terms sharing indices generally do not commute, so the full q-UCCSD expansion retains ordering-dependent operator products.The appendix explicitly identifies nonzero commutators among overlapping single and double excitation operators.

Appendix B: Definition of the gate operations spanning multiple qubits

The appendix defines circuit blocks for operations spanning multiple qubits and explains their decomposition into sequences of local one- and two-qubit operations under nearest-neighbor connectivity.

  • Appendix B: Definition of the gate operations spanning multiple qubits: A spanning one-qubit operation is decomposed into one-qubit operations between the starting qubit q_i and final qubit q_i+N.Each operation in the sequence uses a different set of angles.
  • Appendix B: Definition of the gate operations spanning multiple qubits: The same decomposition principle applies to spanning two-qubit operations, using an operator U_OP selected from the paper’s defined operators.The construction assumes nearest-neighbor connectivity.
  • Appendix B: Definition of the gate operations spanning multiple qubits: Dashed boxes in the circuit figures denote composite one- and two-qubit gate blocks spanning multiple qubits.These blocks are the objects decomposed in the appendix.

Appendix C: Decomposition of the exchange gates in elementary gates

Appendix C describes how the exchange gates U1,ex and U2,ex are implemented and decomposed into elementary gates, with the decomposition summarized in Fig. 9.

  • U1,ex and U2,ex can be implemented directly in a single step using the approach outlined in references [51].The appendix also presents an elementary-gate decomposition to emphasize the resulting gate-count gain.
  • Fig. 9 decomposes an exchange gate Ui,ex for i = 1, 2 between qubits n and m into elementary gates.
  • The appendix separately introduces the elementary-gate forms associated with U1,ex and the UA, UB, and UC gates.
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