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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
TL;DR
The paper addresses scalable quantum simulation of molecular energies without exponentially costly precompilation by experimentally comparing VQE with PEA for molecular hydrogen. VQE achieves chemical accuracy and shows robustness to certain errors, whereas the single-step PEA implementation does not.
Problem
The paper targets electronic structure calculation on quantum hardware without exponentially costly precompilation, an important requirement for scalable molecular simulation.
Method
The authors use superconducting qubits to compute molecular hydrogen’s potential energy curve with UCC-VQE and Trotterized quantum phase estimation.
Results
VQE predicts the dissociation energy within chemical accuracy, while PEA with one Trotter step has an error of (1±1)×10^-2 Hartree.
Takeaways & Limitations
The comparison provides experimental evidence that adaptive VQE can be more resilient to certain errors than traditional gate-model PEA.
Takeaways & Limitations
Because Trotterization is costly, the PEA experiment uses only one Trotter step, which is insufficient for chemical accuracy.
Abstract
from arXiv · showhide
We report the first electronic structure calculation performed on a quantum computer without exponentially costly precompilation. We use a programmable array of superconducting qubits to compute the energy surface of molecular hydrogen using two distinct quantum algorithms. First, we experimentally execute the unitary coupled cluster method using the variational quantum eigensolver. Our efficient implementation predicts the correct dissociation energy to within chemical accuracy of the numerically exact result. Second, we experimentally demonstrate the canonical quantum algorithm for chemistry, which consists of Trotterization and quantum phase estimation. We compare the experimental performance of these approaches to show clear evidence that the variational quantum eigensolver is robust to certain errors. This error tolerance inspires hope that variational quantum simulations of classically intractable molecules may be viable in the near future.
I. VARIATIONAL QUANTUM EIGENSOLVER
VQE uses a parameterized quantum-circuit ansatz and classical minimization to estimate molecular ground-state energies. In molecular hydrogen, the unitary coupled cluster implementation predicts dissociation within chemical accuracy and shows robustness to systematic errors.
- I. VARIATIONAL QUANTUM EIGENSOLVER: The energy surface is measured as a function of circuit parameter θ and bond length R, with local optimization at each bond length.The experiment scanned 1,000 θ values to define the expectation values used for the surface.
- I. VARIATIONAL QUANTUM EIGENSOLVER: VQE parameterizes a quantum state with a circuit and estimates its energy by measuring local Hamiltonian terms.A classical optimizer updates the parameters until the measured energy converges.
- I. VARIATIONAL QUANTUM EIGENSOLVER: The unitary coupled cluster ansatz is used because it can represent states that are not efficiently simulated classically.The ansatz is implemented as a concatenation of parameterized quantum gates.
- I. VARIATIONAL QUANTUM EIGENSOLVER: VQE remains substantially robust to systematic errors because experimentally optimal parameters outperform theoretically optimal parameters.Using theoretically optimal parameters instead gives a dissociation-energy error of 1.1 × 10^-2 Hartree.
II. PHASE ESTIMATION ALGORITHM
The phase-estimation experiment implements the canonical chemistry algorithm by preparing a Hartree–Fock state, applying controlled Trotterized evolution, and measuring phases iteratively. Its long coherent circuits and single Trotter step limit the achieved accuracy, although the complete algorithm is demonstrated experimentally.
- II. PHASE ESTIMATION ALGORITHM: The algorithm prepares a Hartree–Fock state and evolves it under a Hamiltonian using a Trotterized time-evolution operator.The evolution is controlled by an ancilla prepared in a superposition.
- II. PHASE ESTIMATION ALGORITHM: The controlled Trotterized evolution entangles the ancilla and system, allowing measured ancilla phase to identify Hamiltonian eigenvalues.The system collapses to eigenstate |n⟩ with probability |a_n|^2.
- II. PHASE ESTIMATION ALGORITHM: Iterative phase estimation measures the accumulated phase one bit at a time using classical feedback from prior measurements.A majority-voting scheme determines each bit from repeated ancilla measurements.
- II. PHASE ESTIMATION ALGORITHM: A single Trotter step is insufficient for chemical accuracy, and the PEA dissociation-energy error is (1 ± 1) × 10^-2 Hartree.The controlled evolution requires complex circuitry and long coherent evolutions.
- II. PHASE ESTIMATION ALGORITHM: The experiment executes the canonical chemistry algorithm in its entirety, but its protocol benefits from classically optimized Trotter sequences.That classical optimization is intractable for larger molecules.
III. EXPERIMENTAL METHODS
Both algorithms run on a programmable superconducting Xmon-qubit system. The experiments use controlled-phase gates, with PEA requiring substantially more gates and variable-phase operations than VQE.
- III. EXPERIMENTAL METHODS: The experiments use Xmon transmon qubits in a dilution refrigerator at a base temperature of 20 mK.The qubits have tunable frequencies up to 6 GHz.
- III. EXPERIMENTAL METHODS: The entangling operation is a controlled-phase CZφ gate implemented by tuning one qubit near the |11⟩–|02⟩ avoided crossing.Its practical phase range is approximately 0.25 to 5.0 rad.
- III. EXPERIMENTAL METHODS: A VQE sequence uses 11 single-qubit gates and two CZπ gates, whereas a PEA sequence uses at least 51 single-qubit gates, four variable-phase gates, and ten CZπ gates.Additional gates are required when a desired phase cannot be implemented in one physical gate.
IV. CONCLUSION
The experiments compute molecular hydrogen’s potential energy curve with both PEA and VQE. VQE reaches chemical accuracy and is more robust to certain errors, whereas PEA is limited by costly Trotterization.
- The experiments compute molecular hydrogen’s potential energy curve using both quantum phase estimation and the variational quantum eigensolver.
- A single Trotter step prevents the PEA experiment from achieving chemical accuracy.The reported PEA dissociation-energy error is (1±1)×10^-2 Hartree.
- VQE achieves chemical accuracy and shows significant robustness to certain types of error.
- The comparison suggests adaptive algorithms such as VQE may be more resilient than traditional gate-model algorithms such as PEA before error correction.
- The VQE outer loop reduces systematic control errors by optimizing experimentally observed energies without assuming the implemented circuit is ideal.
Appendix A: The electronic structure problem
The appendix develops the electronic-structure Hamiltonian and its qubit representation for molecular hydrogen. Symmetries reduce the Bravyi–Kitaev Hamiltonian to an effective two-qubit problem used in the experiments.
- The electronic structure problem: The central quantum-chemistry problem is finding the lowest energy eigenvalue of the molecular electronic-structure Hamiltonian.The ground state is especially relevant because its energy gap from the first excited state often exceeds room-temperature thermal energy.
- The electronic structure problem: The second-quantized formulation fixes nuclei, represents the wavefunction in a basis, and enforces fermionic antisymmetry.
- The electronic structure problem: The second-quantized Hamiltonian is mapped into qubits using transformations such as Jordan–Wigner or Bravyi–Kitaev.
- The electronic structure problem: For molecular hydrogen in the minimal STO-6G basis, the Bravyi–Kitaev transformation yields a spin Hamiltonian expressed as Pauli-string terms.
- The electronic structure problem: Hamiltonian symmetries stabilize two qubits from the Hartree–Fock starting state, enabling a scalable reduction to an effective two-qubit Hamiltonian.
Appendix B: Experimental methods for VQE
The VQE experiment prepares a unitary coupled-cluster state, measures Hamiltonian expectation values, and minimizes energy over circuit parameters and molecular geometries. The resulting energy surface is compared with theoretical minima and quantified using error estimates.
- Experimental implementation: The VQE experiment uses two active qubits while other qubits are detuned, with calibrated single-qubit and controlled-Z pulse operations.
- Experimental implementation: Expectation values are measured with partial tomography, and chemistry measurements scale polynomially in the number of measurements.
- Experimental implementation: The unitary coupled-cluster pulse sequence is implemented in different Bravyi–Kitaev gauges corresponding to alternative qubit encodings of orbital occupation and parity.
- Energy-surface construction: Errors in the VQE energy surface are greatest where the energy is most sensitive to bond length and rotation angle, while the theoretical minimum curve agrees well with the data.
- Energy-surface construction: The energy at each nuclear separation is obtained by evaluating the Hamiltonian for each θ and selecting the smallest measured energy to construct the energy surface.
Appendix C: Experimental methods for PEA
The PEA experiment uses a three-qubit superconducting circuit with calibrated controlled-phase operations. Decoupling, detuning, and compensating phases are tuned to reduce phase errors, which are critical for phase estimation.
- Experimental implementation: The PEA experiment uses three qubits: one ancilla and two register qubits.
- Experimental implementation: Controlled-Z gates are calibrated with decoupling and detuning strategies that minimize unwanted state-dependent and single-qubit phases.
- Experimental implementation: The phase of the ancilla qubit is the critical parameter targeted by the combined decoupling methods.
- Experimental implementation: Because controlled-phase gates sweep a qubit frequency across a wide range, some phase values require individually tuned compensating phases.
- Experimental implementation: The reported qubit coherence parameters include T1 values of 48.1, 23.7, and 43.0 µs for the three qubits.
CNOT
The CNOT operation is implemented using a CZπ gate together with two single-qubit rotations.
- A CNOT is realized as a CZπ gate plus two rotations.
- The construction replaces the logical CNOT with native controlled-phase and rotation operations.
- This implementation uses three pulse-level components: one CZπ and two rotations.
SWAP
The SWAP operation uses three consecutive CZπ gates with intermediate rotations, while related controlled-Pauli evolutions are compiled from basis changes, parity computation, and controlled phases.
- SWAP: A SWAP is implemented as three consecutive CZπ gates with intermediate rotations.
- SWAP: Z0 uses a CZφ gate and a z rotation on the control qubit.
- SWAP: Z1 surrounds the Z0 construction with SWAP gates so the ancilla interacts with the other qubit.
- SWAP: Controlled X0X1 evolution applies basis changes, computes register parity with a CNOT, applies a controlled phase, then reverses those operations.
- SWAP: Controlled Y0Y1 evolution follows the X0X1 construction with a different basis change.
Appendix D: Unitary coupled cluster
The appendix describes the unitary coupled cluster ansatz used with VQE, its Hartree–Fock reference, truncated UCCSD form, and perturbative initialization for molecular hydrogen.
- Unitary coupled cluster: The UCC ansatz is a unitary variant of coupled cluster with single and double excitations and perturbative triples.
- Unitary coupled cluster: The ansatz is defined relative to a reference state, chosen here as the Hartree–Fock state.
- Unitary coupled cluster: UCCSD retains only the first two terms in the cluster expansion, corresponding to single and double excitations.
- Unitary coupled cluster: VQE determines one- and two-body cluster amplitudes by minimizing a nonlinear variational function.
- Unitary coupled cluster: For molecular hydrogen in the minimal basis, the UCCSD ansatz contains exactly one term.
- Unitary coupled cluster: MP2 or CCSD amplitudes can initialize the optimization, but perturbative single-reference guesses may be poor for strongly multireference or entangled systems.Accurate perturbative estimates can also justify discarding operations associated with very small amplitudes.