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Hartree-Fock on a superconducting qubit quantum computer

Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Sergio Boixo, Michael Broughton, Bob B. Buckley, David A. Buell, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Benjamin Chiaro, Roberto Collins, William Courtney, Sean Demura, Andrew Dunsworth, Daniel Eppens, Edward Farhi, Austin Fowler, Brooks Foxen, Craig Gidney, Marissa Giustina, Rob Graff, Steve Habegger, Matthew P. Harrigan, Alan Ho, Sabrina Hong, Trent Huang, William J. Huggins, Lev Ioffe, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Seon Kim, Paul V. Klimov, Alexander Korotkov, Fedor Kostritsa, David Landhuis, Pavel Laptev, Mike Lindmark, Erik Lucero, Orion Martin, John M. Martinis, Jarrod R. McClean, Matt McEwen, Anthony Megrant, Xiao Mi, Masoud Mohseni, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Hartmut Neven, Murphy Yuezhen Niu, Thomas E. O'Brien, Eric Ostby, Andre Petukhov, Harald Putterman, Chris Quintana, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Doug Strain, Kevin J. Sung, Marco Szalay, Tyler Y. Takeshita, Amit Vainsencher, Theodore White, Nathan Wiebe, Z. Jamie Yao, Ping Yeh, Adam Zalcman

arXiv:2004.04174v4quant-phphysics.chem-ph

TL;DR

Near-term quantum chemistry simulations need scalable variational procedures and practical ways to assess and mitigate device errors. This paper implements Hartree–Fock basis rotations with Givens-rotation circuits, benchmarks them on hydrogen chains and diazene, and applies N-representability-based error mitigation. The experiments reach a dozen qubits and report encouraging accuracy and mitigation effectiveness.

  • Problem

    Near-term quantum devices require benchmarks of fermionic simulation primitives that are relevant to chemistry but generate highly entangled states.

  • Method

    The paper variationally prepares Hartree–Fock states with exactly compiled Givens-rotation basis-change circuits and uses 1-RDM-based fidelity and error-mitigation procedures.

  • Results

    The experiments simulate hydrogen chains through H12 and diazene, with post-selection and purification used to improve measured results.

  • Takeaways & Limitations

    The reported accuracy and error-mitigation effectiveness provide an encouraging signal for progress toward larger quantum simulations.

Abstract

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As the search continues for useful applications of noisy intermediate scale quantum devices, variational simulations of fermionic systems remain one of the most promising directions. Here, we perform a series of quantum simulations of chemistry the largest of which involved a dozen qubits, 78 two-qubit gates, and 114 one-qubit gates. We model the binding energy of ${\rm H}_6$, ${\rm H}_8$, ${\rm H}_{10}$ and ${\rm H}_{12}$ chains as well as the isomerization of diazene. We also demonstrate error-mitigation strategies based on $N$-representability which dramatically improve the effective fidelity of our experiments. Our parameterized ansatz circuits realize the Givens rotation approach to non-interacting fermion evolution, which we variationally optimize to prepare the Hartree-Fock wavefunction. This ubiquitous algorithmic primitive corresponds to a rotation of the orbital basis and is required by many proposals for correlated simulations of molecules and Hubbard models. Because non-interacting fermion evolutions are classically tractable to simulate, yet still generate highly entangled states over the computational basis, we use these experiments to benchmark the performance of our hardware while establishing a foundation for scaling up more complex correlated quantum simulations of chemistry.

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The paper formulates Hartree–Fock as a variational optimization over one-body orbital rotations, connecting Slater-determinant wavefunctions, reduced density matrices, and energy evaluation. The experiments use this framework to demonstrate quantum preparation and conclude that the accuracy and error mitigation are encouraging.

  • Conclusion: The paper concludes that the experiments’ accuracy and error-mitigation effectiveness are encouraging for progress toward larger quantum simulations.This is the stated conclusion of the section.
  • Hartree-Fock formulation: Hartree–Fock seeks the lowest-energy antisymmetrized product of one-particle orbitals, represented as a Slater determinant.The variational principle makes the energy stationary under first-order wavefunction changes.
  • Hartree-Fock formulation: One-body fermionic generators form a closed Lie algebra, so orbital changes can be implemented as unitary basis rotations acting on an initial product state.Thouless’s construction expresses non-orthogonal product wavefunctions through exponentials of one-body fermionic operators.
  • Energy evaluation: The energy-evaluation workflow measures the full 1-RDM, constructs the 2-RDM, and then evaluates the energy.This reduction relies on the Slater-determinant structure of the Hartree–Fock ansatz.
  • Energy evaluation: The rotated state’s 1-RDM determines its energy and gradients with respect to the orbital-rotation parameters.For Slater determinants, the 2-RDM can be constructed directly from the 1-RDM, allowing energy evaluation from measured one-body information.

Appendix B: Implementing the Basis Change Circuit and Circuit Concatenation

The basis-change ansatz implements one-body fermionic evolution exactly with Givens rotations, while circuit structure and measurement schemes are tailored to particle number and orbital filling. These constructions support compact preparation and simultaneous 1-RDM estimation.

  • Implementing the Basis Change Circuit and Circuit Concatenation: Givens rotations implement arbitrary basis rotations exactly, avoiding Trotter error for the one-body component.The rotation sequence is obtained from a QR decomposition of the single-particle unitary.
  • Implementing the Basis Change Circuit and Circuit Concatenation: The compilation exploits fixed particle number by eliminating redundant occupied–occupied rotations and retaining occupied–virtual parameters.The number of non-redundant parameters equals the number of occupied spatial orbitals times the number of virtual orbitals.
  • Circuit structure: The H8 half-filling circuit has a symmetric basis-rotation structure, whereas diazene away from half filling has a parallelogram structure.Diazene contains 8 electrons in 12 orbitals before the two lowest-energy orbitals are frozen.
  • 1-RDM measurement: All diagonal 1-RDM elements are obtained simultaneously by measuring Z expectations on every qubit, with particle-number post-selection available in the computational basis.The diagonal elements equal the probabilities of measuring a 1 on the corresponding qubits.
  • 1-RDM measurement: Two circuits measure all one-off-diagonal 1-RDM elements through X- and Y-basis rotations, but those rotations do not preserve particle number.The circuits separate even and odd pairs and therefore cannot use total-particle-number post-selection.

3. General off-diagonal terms and virtual swapping

Virtual mode relabeling converts non-neighboring fermionic correlators into nearest-neighbor measurements without changing the basis-rotation circuit structure. The resulting strategy reduces the number of measurement settings while preserving particle-number post-selection in specialized circuits.

  • 3. General off-diagonal terms and virtual swapping: Relabeling fermionic modes maps different off-diagonal 1-RDM elements onto nearest-neighbor qubit pairs.Each relabeling changes the Givens rotation angles but preserves the circuit structure.
  • 3. General off-diagonal terms and virtual swapping: Virtual fermionic swaps implement new mode orderings by alternating swaps on even and odd adjacent pairs.For six modes, the example ordering becomes {1, 3, 0, 5, 2, 4}.
  • 3. General off-diagonal terms and virtual swapping: The permutation can be absorbed into the basis rotation at no extra cost by shuffling the unitary’s columns or relabeling its rows and columns.The same relabeling logic can extend to k-RDM elements.
  • Measurement reduction: 13 measurement circuits suffice for the 12-qubit 1-RDM strategy, compared with 276 ungrouped or 149 greedily grouped Pauli measurement circuits.The general scheme requires N/2 circuits, each measured in two or three ways.
  • Particle-number-preserving measurement: Number-preserving basis rotations diagonalize the XX + YY Hamiltonian, enabling computational-basis measurement and particle-number post-selection.The measurement circuit applies to non-overlapping even or odd qubit pairs.

5. Computing error bars for elements of the 1-RDM

The appendix estimates 1-RDM uncertainty by modeling covariances and resampling corrected density matrices, while fidelity witnesses and purification impose physical structure. These procedures support fidelity and energy estimates but depend on assumptions about the state and purification behavior.

  • 5. Computing error bars for elements of the 1-RDM: Covariances between simultaneously measured 1-RDM elements are estimated from qubit-observable covariances.The resulting covariance matrices are constructed for each circuit permutation.
  • 5. Computing error bars for elements of the 1-RDM: A second covariance procedure assumes sampling from a pure Gaussian state, for which the 2-RDM is exactly determined by the 1-RDM.This approximation is considered applicable when fidelity is sufficiently high because covariance changes are second order.
  • 5. Computing error bars for elements of the 1-RDM: The analysis resamples the 1-RDM 1000 times, then computes means and standard deviations after enforcing positivity and the correct trace.A fixed-trace positive projection is applied to ensure each resampled 1-RDM is positive semidefinite.
  • Fidelity estimation: For non-interacting fermion circuits, a fidelity witness gives a strict lower bound using only L^2 expectation values.The witness can be evaluated from the measured 1-RDM.
  • Purification and fidelity: McWeeny purification iteratively projects a measured 1-RDM toward the set of idempotent matrices before fidelity and energy evaluation.The fidelity procedure diagonalizes the purified 1-RDM and compares basis-rotation unitaries through their determinant overlap.

Appendix E: Error mitigation through purification

The appendix analyzes McWeeny purification as a projection of sampled 1-RDMs onto the idempotent, fixed-trace, positive-semidefinite projector manifold. It establishes convergence under sampling noise, with conditions depending on error moments and distributional assumptions.

  • Purification method: McWeeny purification iteratively moves a measured 1-RDM toward an idempotent projector while preserving the target Slater-determinant structure.The direct optimization imposes fixed trace, positive semidefiniteness, and projector constraints, whereas the iterative method is practical under mild conditions.
  • Convergence analysis: The eigenvalue variance converges quadratically, requiring K ∈ O(log log(1/ϵ)) iterations when the convergence criteria are met.The stated criterion is 9σ4(α4 −1) + 4|σ|5(α6 + 3α5 + 3α3) ≤ σ2.
  • Convergence analysis: Under Gaussian sampling assumptions, McWeeny purification converges when the noise variance satisfies a sufficient moment-dependent criterion.The criterion uses α4, α5, and α6 to bound the recurrence for the principal eigenvalue variance.
  • Convergence analysis: For non-Gaussian sampling noise, convergence follows when the distribution moments are appropriately small.The appendix notes that Eq. (E9) can be used to establish convergence beyond the Gaussian case.

1. Errors in Eigenvalues

This section bounds errors in reconstructed 1-RDM eigenvalues using spectral-norm perturbation results and models the 1-RDM’s independent matrix elements under Gaussian noise.

  • Perturbation model: Eigenvalue errors are bounded using the spectral norm of the perturbation matrix added to the true density operator.The bound is stated for ˜ρ = ρ + sE with s ∈ [0, 1].
  • Noise model: The error matrix is modeled with M independently distributed, zero-mean elements whose variances are at most σ2.This stochastic model supports analytical bounds on reconstructed eigenvalue errors.
  • Convergence condition: Under Gaussian assumptions, the resulting eigenvalue-error bounds yield a convergence condition for the purification analysis.The appendix states that the upper bounds can be computed from the matrix-element model under Gaussianity.
  • Noise model: The 1-RDM contains M = N(N + 1)/2 independent matrix elements in this analysis.This relation determines how system size enters the sampling and convergence bounds.

2. Errors in Eigenvectors

This section addresses whether purification converges to the correct pure state, not merely whether its eigenvalue variance decreases. The analysis uses eigenvalue gaps and perturbation theory to bound eigenvector errors.

  • Scope of the analysis: Purification can converge while still producing the wrong pure state if noise causes an eigenvalue crossing.The relevant issue is whether the converged state remains ϵ-close to the true state.
  • Perturbation analysis: The eigenvector analysis interpolates between the true density operator and its perturbed version using R time slices.At slice j, the operator is ρ(j) := ρ + (j/R)E.
  • Perturbation analysis: First-order perturbation theory bounds successive eigenvector changes assuming a nonzero eigenvalue gap.The bound is expressed using the minimum gap γ(j) between the principal eigenvalue and the rest of the spectrum.
  • Perturbation analysis: The analysis combines bounds across slices with the triangle inequality to control the Euclidean distance between endpoint eigenvectors.The resulting distance estimate is derived from the successive-step perturbation bounds.
  • Assumptions: Bounding the minimum eigenvalue gap requires the assumption ∥E∥ ≤ 1/4 and additional perturbative control of the denominator.The appendix explicitly notes that γmin is not known a priori.
  • Sampling requirements: Even with Chebyshev’s inequality, the failure probability is asymptotically negligible when σM is chosen according to Eq. (E25).Because M ∈ Θ(N2), the required sample count is stated as Θ(ϵ/N).

Appendix F: Effect of CPHASE and Givens Rotation Error

The appendix models coherent cphase and stochastic Givens-rotation errors in the basis-rotation circuits, then evaluates local phase corrections, VQE, and purification-based mitigation.

  • Error models: The analysis benchmarks two gate errors occurring in Givens-rotation circuits using analytical and numerical models.The errors include an always-on cphase interaction and stochastic Rz control-angle errors.
  • Error models: The always-on cphase(π/24) has negligible effect on the experiment, whereas stochastic Rz(θ) errors coherently corrupt circuit outputs.The cphase error is modeled as a |11⟩⟨11| phase applied after the iswap gate.
  • cphase mitigation: Local Rz gates are used to counteract the parasitic cphase(π/24) in the diazene simulations.The corrected circuits are compared with the original cphase circuits and with VQE optimization.
  • cphase mitigation: For iswap φ = π/24, the expected errors are Err1 ≈ 0.32% and Err2 ≈ 0.11%.The improvement is reported as most beneficial for in-plane diazene rotations, especially near transition states where sensitivity is higher.
  • Stochastic Rz errors: The stochastic Rz model propagates control-angle noise into the 1-RDM through a map derived from the noisy Givens rotation.The model assumes stochasticity occurs on a timescale faster than a single energy evaluation.
  • Stochastic Rz errors: When σ > 0.22, purification projects to the wrong 1-RDM under the modeled stochastic Rz noise.This threshold marks a failure regime for the purification-based correction in the stated model.

Appendix G: Gradient for the Basis Rotation Ansatz

The appendix derives gradients for the basis-rotation ansatz using one-body operators and 1-RDM-accessible quantities. It also describes an iterative, regularized optimization procedure for updating the Slater-determinant wavefunction.

  • The ansatz gradient can be evaluated from the 1-RDM when the state is a Slater determinant.
  • Nested commutators from the Baker-Campbell-Hausdorff expansion are evaluated through adjoint actions of the generator.
  • Diagonalizing the N × N coefficient matrix avoids diagonalizing a larger 2n × 2n matrix when evaluating the adjoint action.
  • The gradient operator ∇f(Z) is a one-body operator distinct from the rotation generator associated with c_b,i.
  • The optimizer updates κ using an augmented-Hessian procedure with level shifting and regularization of oversized parameter updates.
  • Each update remains a non-interacting fermion wavefunction, and iterations stop after a fixed number of steps or when the commutator falls below a threshold.

1. Post-selection performance

Post-selection performance was assessed through retained circuit repetitions, readout distributions, and natural occupation numbers across hydrogen-chain and diazene experiments. The retained fraction decreased with system size while tracking approximately 95% joint readout fidelity.

  • The retained fraction of circuit repetitions decreases with system size and almost perfectly tracks a joint readout fidelity of 95%.
  • Post-selection improves the scatter of local qubit occupation probabilities toward the theoretical 1-RDM values.
  • The readout analysis compares raw and post-selection data through natural occupation numbers.

3. Energy and Fidelity

Fidelity witnesses correlate with absolute energy error across the experiments, while error mitigation produces consistently high perceived fidelity in the 10-qubit calculations. Performance nevertheless varies across collection days because of slow TLS diffusion.

  • The correlation between fidelity witness and absolute energy error suggests fidelity can serve as an optimization target for basis-rotation benchmarks.
  • Slow TLS diffusion changes qubit performance over hours or days, producing variance across data-collection days.
  • On all but one 10-qubit experiment, variational relaxation and other mitigation techniques achieved > 98.0% average fidelity.

Appendix J: Molecular geometries

The molecular geometries were generated differently for hydrogen chains and diazene. Hydrogen-chain calculations sampled fixed atom separations, whereas diazene geometries followed constrained reaction-coordinate optimizations and subsequent reduction to 10 qubits.

  • Diazene reaction coordinates were generated by optimizing molecular geometries while constraining either the dihedral angle or NNH angle.
  • Two canonical Hartree-Fock self-consistent-field cycles followed by integrating out the bottom two energy levels reduced diazene to a 10-qubit problem.
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