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Low-depth circuit ansatz for preparing correlated fermionic states on a quantum computer

Pierre-Luc Dallaire-Demers, Jonathan Romero, Libor Veis, Sukin Sim, Alán Aspuru-Guzik

arXiv:1801.01053v1quant-ph

TL;DR

Representing interacting-fermion states requires exponentially scaling classical memory, while near-term quantum simulation motivates low-depth state preparation. This work prepares a Gaussian reference with matchgates and adds nearest-neighbor phase coupling; the resulting LDCA is reported to have potential for accurate strongly correlated ground states and pairing-field Hamiltonians.

  • Problem

    Representing many-body fermionic states requires memory that scales exponentially with particle number, motivating quantum simulation.

    Source: §I · p. 1

  • Method

    The paper combines a fermionic Gaussian reference prepared with linear-depth matchgates and nearest-neighbor phase coupling in a low-depth VQE ansatz.

    Source: §I · p. 1, §V · p. 13

  • Results

    LDCA showed potential to accurately describe exact ground states in Fermi-Hubbard clusters and cyclobutadiene, while BUCC and LDCA can approximate pairing-field Hamiltonians.

    Source: §V · p. 13

  • Takeaways & Limitations

    BUCC and LDCA extend VQE to pairing-field problems not addressed by previous ansatzes such as UCC.

    Source: §V · p. 13

    Circuit-depth estimates assume a linear qubit chain and directly implementable nearest-neighbor two-qubit gates; possible improvements from increased connectivity remain open.

    Source: §IV · p. 13

Abstract

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Quantum simulations are bound to be one of the main applications of near-term quantum computers. Quantum chemistry and condensed matter physics are expected to benefit from these technological developments. Several quantum simulation methods are known to prepare a state on a quantum computer and measure the desired observables. The most resource economic procedure is the variational quantum eigensolver (VQE), which has traditionally employed unitary coupled cluster as the ansatz to approximate ground states of many-body fermionic Hamiltonians. A significant caveat of the method is that the initial state of the procedure is a single reference product state with no entanglement extracted from a classical Hartree-Fock calculation. In this work, we propose to improve the method by initializing the algorithm with a more general fermionic Gaussian state, an idea borrowed from the field of nuclear physics. We show how this Gaussian reference state can be prepared with a linear-depth circuit of quantum matchgates. By augmenting the set of available gates with nearest-neighbor phase coupling, we generate a low-depth circuit ansatz that can accurately prepare the ground state of correlated fermionic systems. This extends the range of applicability of the VQE to systems with strong pairing correlations such as superconductors, atomic nuclei, and topological materials.

I. INTRODUCTION

The paper proposes a low-depth VQE ansatz built around fermionic Gaussian reference states to address correlated-fermion ground states. Its introduction motivates quantum simulation by the exponential memory required for classical many-body state representations and outlines the theoretical framework behind the approach.

I. INTRODUCTION

  • Representing many-body fermionic states requires memory that scales exponentially with particle number, motivating quantum simulation.The introduction also cites limited coherence and finite experimental precision as reasons to seek low-depth circuits.Source: §I · p. 1
  • The low-depth circuit ansatz (LDCA) combines linear-depth matchgate blocks with nearest-neighbor phase coupling to prepare approximate ground states of fermionic Hamiltonians.The paper benchmarks it on Fermi-Hubbard clusters and cyclobutadiene automerization.Source: §I · p. 1

II. GENERALIZED VARIATIONAL QUANTUM EIGENSOLVER

  • The section develops BUCC and generalized Hartree-Fock foundations, then covers Gaussian-state preparation, LDCA, and analytical gradient evaluation.It also previews numerical benchmarks on the Fermi-Hubbard model and cyclobutadiene.Source: §II · pp. 1–2, §I · p. 1

A. Formulation of the problem

  • The ground-state problem is framed as minimizing energy over states, with variational ansatzes used when analytic or numerically exact minimization is unavailable.The cited section introduces Bogoliubov unitary coupled cluster (BUCC) as one such ansatz.Source: A. Formulation of the problem · p. 2

B. Bogoliubov unitary coupled cluster theory

  • Bogoliubov- and quasiparticle-based coupled-cluster methods extend coupled-cluster methods to systems with mean-field paired states.The paper presents the unitary version in preparation for quantum-computer implementation.Source: B. Bogoliubov unitary coupled cluster… · p. 2
  • The BUCC ansatz applies a unitary transformation to a reference state, remains normalized, and can be truncated at single or double excitations.Because quasiparticle operators generally do not conserve particle number, optimization imposes a particle-number constraint.Source: B. Bogoliubov unitary coupled cluster… · p. 3

C. Generalized Hartree-Fock theory

The generalized Hartree-Fock reference is represented as a fermionic Gaussian state and obtained through its covariance matrix. This framework includes both Slater determinants and mean-field BCS states.

C. Generalized Hartree-Fock theory

  • Fermionic Gaussian states include Hartree-Fock Slater determinants and mean-field BCS states, and provide a reference for quantum simulations.The method obtains the Bogoliubov matrix from the covariance matrix without a self-consistent loop.Source: C. Generalized Hartree-Fock theory · p. 3
  • A Gaussian state is characterized by a real antisymmetric covariance matrix; for a pure state, its square equals minus the identity.The covariance matrix is defined through expectations of Majorana-operator commutators.Source: C. Generalized Hartree-Fock theory · p. 3
  • The covariance-matrix formalism is used to recover the Bogoliubov transformation that defines the reference state.The paper recasts the Gaussian state using a single-particle density matrix.Source: C. Generalized Hartree-Fock theory · p. 3

1. Finding the ground state

The ground-state procedure uses Gaussian-state expectations to construct a state-dependent quadratic Hamiltonian and evolve the covariance matrix. Imaginary-time and fixed-point steps provide complementary routes to the reference state.

1. Finding the ground state

  • Wick’s theorem expresses Gaussian-state expectation values through covariance-matrix submatrices and yields an effective state-dependent quadratic Hamiltonian.The imaginary-time evolution lowers energy and preserves purity, but may become trapped in a local minimum.Source: §1 · p. 4
  • A complementary approach minimizes the free energy through fixed-point iterations of a transcendental equation.The numerical experiments apply this after imaginary-time evolution.Source: §1 · p. 4
  • Imaginary-time evolution followed by fixed-point evolution was numerically stable and consistently reached the desired generalized Hartree-Fock ground state.The passage identifies this sequence as the numerical procedure used in the experiments.Source: §1 · p. 4

D. The quantum subroutine

The quantum subroutine prepares Gaussian references with matchgates and extends the circuit with non-matchgate variational terms. The resulting LDCA uses nearest-neighbor phase coupling to build a low-depth VQE ansatz.

D. The quantum subroutine

  • A pure Gaussian reference can be prepared on a quantum register with a linear-depth algorithm using a fixed circuit structure with variable parameters.The paper distinguishes this parametrized structure from the gate-decomposition strategy described in [51].Source: D. The quantum subroutine · p. 4

1. Matchgate decomposition of a Bogoliubov transformation

  • The Bogoliubov transformation can be implemented with nearest-neighbor matchgates, decomposed into SO(4) pair rotations and SO(2) local phases.The decomposition uses parallel operations in a linear qubit chain to implement the transformation.Source: §1 · p. 5

D. The quantum subroutine

  • The LDCA adds variational non-matchgate terms to a circuit sequence resembling the Bogoliubov transformation decomposition.The broader subroutine is designed to represent BCS-like pairing correlations.Source: D. The quantum subroutine · p. 4, §1 · p. 5

1. Matchgate decomposition of a Bogoliubov transformation

  • The circuit interleaves matchgates on even and odd nearest-neighbor pairs, with local phase rotations completing the Bogoliubov circuit.Figure 1 illustrates the decomposition on eight qubits.Source: §1 · pp. 5–6, Figure 1 · p. 6

2. A low-depth circuit ansatz

The LDCA augments matchgate circuits with nearest-neighbor phase coupling to target correlated fermionic states, starting from a prepared quasiparticle vacuum. Its circuit depth grows linearly with qubit count, and accuracy can be improved by adding cycles.

2. A low-depth circuit ansatz

  • The quasiparticle-basis VQE can represent nonnumber-conserving terms such as pairing fields, but the basis change may increase the number of measurements.The described route applies the inverse Bogoliubov transformation before measuring the Hamiltonian and number operator in the original orbital basis.Source: §2 · p. 7
  • BUCCSD-like parameter scaling is expected to be impractical under near-term processor error rates.The discussion assumes particle number proportional to orbital count.Source: §2 · p. 7
  • Nearest-neighbor σz⊗σz rotations augment matchgates to mimic quartic T2 effects in a composable ansatz for correlated fermionic states.The protocol prepares the quasiparticle vacuum in the Bogoliubov picture and measures in the original basis after an inverse Bogoliubov transformation.Source: §2 · p. 7
  • Circuit depth is (10L + 8) + 4 and linear in qubit count; adding cycles can improve accuracy until convergence or errors limit precision.The depth count includes the inverse Bogoliubov transformation and initial single-qubit X gates.Source: §2 · p. 8
  • The paper outlines a quantum-resource implementation of analytical LDCA gradients to guide VQE optimization toward the ground state and its energy.Source: §2 · p. 8

E. Gradient Evaluation for LDCA

The section develops a quantum-circuit procedure for evaluating analytical LDCA energy gradients. It also describes measurement and optimization trade-offs and reports that LDCA prepares exact ground states in the tested systems.

E. Gradient Evaluation for LDCA

III. NUMERICAL EXAMPLES

In 8-qubit tests of the half-filled Fermi-Hubbard model and cyclobutadiene, two-cycle LDCA exactly recovered the ground states. One-cycle LDCA was less accurate than BUCCSD, while requiring shallower circuits than the cited higher-order BUCC approach.

III. NUMERICAL EXAMPLES

  • The tests compare GHF, BUCCSD, and one- and two-cycle LDCA on 8-qubit half-filled Fermi-Hubbard and cyclobutadiene PPP Hamiltonians.Source: §III · p. 10
  • Two-cycle LDCA exactly recovers the simulated ground states; one-cycle LDCA outperforms GHF but is less accurate than BUCCSD.Source: §III · p. 10
  • One-cycle LDCA adds 41 depth for total depth 75 and 148 parameters; two-cycle LDCA adds 81 depth for total depth 115 and 288 parameters.Source: §III · p. 10
  • Two-cycle LDCA uses 288 parameters, exceeding the 256-dimensional Hilbert space, but its depth is shorter than BUCC through fourth order, which is required for exact recovery in these systems.Source: §III · p. 10

A. Fermi-Hubbard model

On an 8-qubit, half-filled 2 × 2 Fermi-Hubbard cluster, 2-cycle LDCA reaches the exact ground state across interaction strengths, while fewer-cycle and other ansatzes are less accurate. The section also reports exact preparation by BUCCSD and 1-cycle LDCA for a 2-site cluster.

A. Fermi-Hubbard model

  • The 2D Fermi-Hubbard model has a rich phase diagram, but a general exact solution remains elusive.The section motivates the model as relevant to understanding high-temperature cuprate superconductors.Source: A. Fermi-Hubbard model · p. 11
  • 2-cycle LDCA reaches the exact ground state up to numerical accuracy across all interaction strengths in the half-filled 2 × 2 cluster.The cluster can be implemented on an 8-qubit processor; 1-cycle LDCA is intermediate between GHF and BUCCSD, and BUCCSD does not reach the exact ground state at strong interactions.Source: A. Fermi-Hubbard model · p. 11
  • For the 2-site one-dimensional cluster, both BUCCSD and 1-cycle LDCA reach the exact ground state for every interaction strength.BUCCSD is equivalent to full configuration interaction in this specific case.Source: A. Fermi-Hubbard model · p. 11
  • The authors cannot determine how the required LDCA cycle count scales with cluster size; resolving it would require more intensive numerics.The 2-site case needs one cycle and the 2 × 2 case reaches the ground state in two, leaving open whether scaling is non-exponential.Source: A. Fermi-Hubbard model · p. 11

B. Cyclobutadiene

For cyclobutadiene automerization, 2-cycle LDCA produces the numerically exact ground state across the reaction path. BUCCSD improves on GHF near the D2h ground state, but its improvement is less pronounced near the strongly correlated D4h transition state.

B. Cyclobutadiene

  • The D4h transition state is strongly correlated, unlike the weakly correlated closed-shell D2h ground state.Accurately treating the transition state is presented as important for assessing proposed reaction-mechanism observations and the automerization barrier.Source: B. Cyclobutadiene · p. 11
  • BUCCSD considerably improves on GHF near the D2h ground state, but the improvement is less prominent near the strongly correlated D4h transition state.The study uses a Pariser-Parr-Pople model, which the authors say captures the main physics of π-electron systems and connects to the Fermi-Hubbard model.Source: B. Cyclobutadiene · p. 11
  • 2-cycle LDCA produces the numerically exact ground state for every value of the reaction coordinate λ.The 1-cycle method has accuracy between GHF and BUCCSD.Source: B. Cyclobutadiene · p. 12

IV. DISCUSSION

The discussion links LDCA’s accuracy and depth to its tunable cycles and pairing terms, while identifying implementation assumptions and open questions. It also points to possible extensions beyond the tested fermionic systems.

IV. DISCUSSION

  • The authors suggest LDCA could outperform BUCC in accuracy and efficiency, while each added cycle increases circuit depth.LDCA with all ZZ angles set to zero recovers BUCC with single excitations; that choice cannot improve GHF because it only rotates a basis already optimized by the Bogoliubov transformation.Source: §IV · p. 12
  • A Bravyi-Kitaev mapping may shorten measured Pauli strings from O(M) to O(log M), and traditional UCCSD gives the same results as BUCCSD for Hamiltonians without explicit pairing terms.The proposed mapping is a possible route to shorter strings, not a demonstrated result.Source: §IV · p. 12
  • Restricting the ansatz to number-conserving terms reduces overlap with the exact Fermi-Hubbard ground state, making pairing-amplitude reconfiguration important for accuracy.The authors note that constraints on LDCA variational parameters might reduce their number, but do not establish such a reduction.Source: §IV · pp. 12–13
  • The proposed depth estimates assume a linear qubit chain and directly implementable nearest-neighbor two-qubit gates.With tunable nearest-neighbor entangling gates, basis rotations suffice to implement the required gates with only a small depth overhead; higher connectivity is left open.Source: §IV · p. 13
  • The authors propose calibrating Bogoliubov-circuit angles against measured energy and particle number, and estimating LDCA errors by comparison with classical values.They also suggest replacing ZZ rotations with equivalent time delays.Source: §IV · p. 13
  • The formalism may extend to nucleon simulation and gauge theories, but the authors did not numerically simulate these systems.Nucleon simulations were left aside as beyond the authors’ expertise.Source: §IV · p. 13

V. CONCLUSION

The paper presents a unitary Bogoliubov coupled-cluster VQE framework with a Gaussian reference prepared by linear-depth matchgates, extended into LDCA using nearest-neighbor ZZ rotations. Tests on Fermi-Hubbard and cyclobutadiene systems show strong accuracy, including pairing-field cases, while broader molecular benchmarking and cycle-scaling remain future work.

V. CONCLUSION

  • Fermionic Gaussian reference states, including Slater determinants and mean-field BCS states, can be prepared with linear-depth nearest-neighbor matchgate circuits.Adding nearest-neighbor σz⊗σz rotations yields LDCA, and each added cycle increases circuit depth linearly.Source: §V · p. 13
  • Tests on Fermi-Hubbard clusters and cyclobutadiene show LDCA can accurately describe exact ground states of strongly correlated fermionic systems, including Hamiltonians with pairing fields.The authors state this pairing-field capability extends VQE’s applicability beyond ansatzes such as UCC.Source: §V · p. 13
  • Accuracy across a broader variety of molecular systems and the scaling of required LDCA cycles for general systems remain future work.BUCC and LDCA do not conserve particle number, so particle-number constraints must be imposed during classical optimization.Source: §V · pp. 13–14

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