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Towards Quantum Chemistry on a Quantum Computer
Benjamin P. Lanyon, James D. Whitfield, Geoff G. Gillet, Michael E. Goggin, Marcelo P. Almeida, Ivan Kassal, Jacob D. Biamonte, Masoud Mohseni, Ben J. Powell, Marco Barbieri, Alán Aspuru-Guzik, Andrew G. White
TL;DR
Large-molecule quantum simulation faces an accurate time-evolution decomposition challenge as direct decompositions become impractical. The paper uses iterative phase estimation with repeated sampling and Hamiltonian simulation methods, achieving high precision while exposing scaling limits from growing gate requirements.
Problem
Accurate decomposition of molecular time-evolution operators becomes impractical for large molecules, although efficient first-principles simulation remains possible.
Method
The approach combines iterative phase estimation, repeated sampling with majority voting, and Trotter-Suzuki approximations for non-commuting Hamiltonian terms.
Results
Near-perfect success probability was achieved up to 47 bits, yielding energy precision of ≈10^-13Eh; precision of ±10^-4Eh required 6 Trotter steps and 522 gates for U^1.
Takeaways & Limitations
The results support quantum-computing approaches to molecular simulation while indicating that larger systems require efficient propagator simulation rather than direct decomposition.
Takeaways & Limitations
For larger implementations, powers of U generally require roughly twice as many gates for each additional precision digit, amplifying gate errors and limiting obtainable precision.
Abstract
from arXiv · showhide
The fundamental problem faced in quantum chemistry is the calculation of molecular properties, which are of practical importance in fields ranging from materials science to biochemistry. Within chemical precision, the total energy of a molecule as well as most other properties, can be calculated by solving the Schrodinger equation. However, the computational resources required to obtain exact solutions on a conventional computer generally increase exponentially with the number of atoms involved. This renders such calculations intractable for all but the smallest of systems. Recently, an efficient algorithm has been proposed enabling a quantum computer to overcome this problem by achieving only a polynomial resource scaling with system size. Such a tool would therefore provide an extremely powerful tool for new science and technology. Here we present a photonic implementation for the smallest problem: obtaining the energies of H2, the hydrogen molecule in a minimal basis. We perform a key algorithmic step - the iterative phase estimation algorithm - in full, achieving a high level of precision and robustness to error. We implement other algorithmic steps with assistance from a classical computer and explain how this non-scalable approach could be avoided. Finally, we provide new theoretical results which lay the foundations for the next generation of simulation experiments using quantum computers. We have made early experimental progress towards the long-term goal of exploiting quantum information to speed up quantum chemistry calculations.
I. METHODS SUMMARY
The H2 calculation reduces full configuration interaction to eigenvalue estimation for two one-qubit Hamiltonian blocks, using a photonic iterative phase-estimation implementation with classically assisted preprocessing.
- Hamiltonian reduction: Finding the eigenvalues of the two block Hamiltonians through phase estimation amounts to performing full configuration interaction.The experiment encodes exact eigenstates from a preliminary classical calculation, while the appendix analyzes robustness to imperfect encoding.
- Classical preprocessing: The demonstration uses a propagator time step of t=1 ℏ/Eh and evaluates the required molecular integrals classically with Hartree-Fock.The resulting Hamiltonian matrix elements are used to construct the unitary operators for the experiment.
- Gate implementation: Each one-qubit evolution operator is decomposed into a global phase and single-qubit rotations, with powered operators obtained by scaling α and γ by j.The angle β remains unchanged when constructing U^j.
A. Minimal basis and symmetries in the electronic Hamiltonian of the hydrogen molecule
The H2 minimal-basis simulation uses six antisymmetric two-electron configurations, whose symmetry separates the Hamiltonian into coupled and uncoupled subspaces.
- Basis construction: The four spin-orbitals are combined antisymmetrically into six two-electron configurations forming the simulation basis.The orbitals arise from bonding and antibonding combinations of the two 1s atomic orbitals, including electron spin.
- Hamiltonian symmetries: Symmetry makes the Hamiltonian block-diagonal across subspaces {|Φ1⟩, |Φ6⟩}, {|Φ2⟩}, {|Φ3⟩, |Φ4⟩}, and {|Φ5⟩}.Most basis states do not mix under the Hamiltonian.
- Spin structure: The states |Φ2⟩, |Φ5⟩, and (|Φ3⟩+|Φ4⟩)/2 form a three-fold-degenerate triplet state with angular momentum S=1.
B. Details of computational methods
The computational methods obtain molecular integrals through restricted Hartree-Fock calculations on a classical computer using the STO-3G basis and PyQuante.
- Classical calculation: Restricted Hartree-Fock calculations were performed classically with the STO-3G basis.
- Software: The calculations used the PyQuante quantum chemistry package version 1.6.
- Hamiltonian construction: The resulting molecular integrals were used to evaluate the matrix elements of H^(1,6) and H^(3,4).
C. Classical error correction technique
Repeated sampling and majority voting provide a classical error-correction technique for improving individual IPEA bit identification when its single-sample success probability exceeds 0.5.
- Error sources: Imperfect gates and phase truncation reduce the probability of correctly identifying an individual bit with one sample.
- Majority voting: When single-sample bit success remains above 0.5, repeated sampling followed by majority voting improves the probability of correct identification.
- Large-scale boundary: Large-scale implementations may require quantum error-correction techniques because numerous circuit errors make maintaining bit success above 0.5 challenging.
D. Count rates
The experiment used a low-brightness photonic source to reduce multi-photon errors, producing about 15 coincident detections per second and enabling repeated iterations.
- Count rates: ≈50 mW pumping power was used to reduce unwanted multi-photon-pair emissions.The detectors could not distinguish these emissions, which introduced circuit-operation errors.
- Count rates: 15 coincident detection events per second were obtained at the optical-circuit output.
- Count rates: Each iterative phase-estimation step could therefore be repeated 15 times per second.
A. Efficient simulation of arbitrary molecular time-evolution operators
The paper develops an efficient molecular-simulation pipeline by expressing chemical Hamiltonians in second-quantized and qubit representations, then decomposing their evolution into implementable components.
- A. Efficient simulation of arbitrary molecular time-evolution operators: Trotter-Suzuki expansion decomposes the full propagator into evolution operators for non-commuting Hamiltonian terms.Each term can then be simulated through corresponding quantum circuits.
- A. Efficient simulation of arbitrary molecular time-evolution operators: The second-quantized chemical Hamiltonian contains O(N 4) terms for N single-electron basis functions.
- A. Efficient simulation of arbitrary molecular time-evolution operators: The Hamiltonian’s annihilation and creation operators obey fermionic anticommutation relations, with indices spanning all N basis functions.The one- and two-electron integrals are evaluated during a preliminary Hartree-Fock procedure.
- A. Efficient simulation of arbitrary molecular time-evolution operators: A system with N single-electron spin-orbitals requires N qubits to represent orbital occupancy.The N-qubit Hilbert space permits states with any number of electrons.
- A. Efficient simulation of arbitrary molecular time-evolution operators: Jordan-Wigner transformations map fermionic creation and annihilation operators into Pauli-spin representations suitable for quantum computers.The mapping preserves occupied and unoccupied orbital states in the qubit computational basis while maintaining antisymmetrization.
Step 3. Exponentiation of the Hamiltonian
The full molecular propagator is approximated by composing evolutions under individual Hamiltonian terms, balancing Trotter error against computational effort and experimental precision.
- Step 3. Exponentiation of the Hamiltonian: Individual Hamiltonian terms can be simulated efficiently, but their noncommutation prevents reconstructing the full propagator from simple direct products.
- Step 3. Exponentiation of the Hamiltonian: Trotter-Suzuki relations approximate the full unitary propagator by composing evolutions of non-commuting operators.
- Step 3. Exponentiation of the Hamiltonian: As the Trotter number tends to infinity, equivalently dt →0, the approximation becomes exact.Practical calculations compromise between computational effort and accuracy.
- Step 3. Exponentiation of the Hamiltonian: The IPEA requires powers U^j of the evolution operator, whose gate counts grow with j and amplify experimental errors.Each additional bit provides an exponential increase in precision, while the required gates increase exponentially with the number of bits.
- Step 3. Exponentiation of the Hamiltonian: Hamiltonians diagonal in the computational basis, such as the classical Ising model, do not require a Trotter expansion for accurate simulation.
Step 4. Circuit representations of the unitary propagator
Circuit decompositions implement exponentiated molecular Hamiltonian terms, while resource estimates show that the H2 demonstration is small but larger simulations retain polynomial scaling.
- Step 4. Circuit representations of the unitary propagator: Analytical gate decompositions provide circuits for implementing exponentiated tensor products of Pauli matrices.The summarized networks realize U(dt) for a general molecular Hamiltonian.
- Step 4. Circuit representations of the unitary propagator: The gate count for simulating each Jordan-Wigner term is linear in the number of intervening qubits.
- Step 4. Circuit representations of the unitary propagator: The estimated gate-scaling for simulating a general many-electron chemical Hamiltonian is O(N 5), excluding noise.
- Step 4. Circuit representations of the unitary propagator: The H2 minimal-basis calculation uses five qubits, including one control qubit for iterative phase estimation.Four register qubits represent the single-electron spin-orbitals.
- Step 4. Circuit representations of the unitary propagator: ±10−4Eh precision is achieved at a Trotter number of 6, corresponding to 522 gates for constructing U^1.This estimate includes one- and two-qubit operations but excludes error correction and assumes a continuous gate set.
- Step 4. Circuit representations of the unitary propagator: Although the estimates exceed current quantum-computer capabilities, the resource requirements grow polynomially with system size.
B. Additional experimental results
Additional experiments show that repeated sampling substantially improves IPEA success probability and that the implementation remains robust to imperfect eigenstate preparation. They also clarify that the demonstrated precision relies on a small-scale implementation strategy that will not generally scale.
- Success probabilities: The probability of correctly identifying each bit decreases with experimental and theoretical errors, but majority voting improves it exponentially with sample count when single-sample success exceeds 0.5.This classical error-correction strategy increases success probability at the cost of repeating the experiment, but large circuits may require quantum error correction.
- Phase precision: Using n = 101 samples per bit, the experiment achieves near-perfect success probability up to 47 extracted bits, corresponding to energy precision of ≈10−13Eh.The reported limit is imposed by machine-level precision in the classical preprocessing of the Hamiltonians.
- Scaling limitation: For larger implementations, each additional precision digit roughly doubles gate requirements and amplifies gate errors because powers of the evolution operator cannot generally be re-encoded directly.The constant per-bit gate-error probability achieved in this small demonstration therefore does not generally persist at scale.
- Eigenstate fidelity: The implementation is robust for encoded-register fidelity F ≳0.5 because repeated sampling can amplify per-bit success probabilities above 0.5 toward unity.For F ≲0.5, measured success probabilities are very low.
- State preparation: With destructive measurement of both control and register qubits after each iteration, the experiment must re-prepare the register state for every IPEA iteration.A sequential scheme could instead use the post-measurement register state as the next iteration’s input, allowing measurement-induced eigenstate purification.
- Probability estimation: The experiment estimates success probabilities by forcing accepted feedforward trajectories and using n = 301 samples to infer majority-vote probabilities without running every trajectory directly.Direct experimental estimation would require over 100 repetitions for a 10% error estimate and more than 100 hours of waveplate rotation for the displayed datasets.