Simulation of Electronic Structure Hamiltonians Using Quantum Computers

James D. Whitfield; Jacob Biamonte; Alán Aspuru-Guzik

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Simulation of Electronic Structure Hamiltonians Using Quantum Computers

James D. Whitfield, Jacob Biamonte, Alán Aspuru-Guzik

arXiv:1001.3855v3Jan 21, 2010quant-phphysics.chem-ph

TL;DR

Classical methods lack computational efficiency for exact quantum-system simulation as molecular descriptions grow. The paper constructs quantum circuits from molecular integrals using Hamiltonian decomposition, Trotterization, Jordan-Wigner mapping, phase estimation, and adiabatic state preparation, and demonstrates the construction for minimal-basis H2.

Problem
Classical numerical methods for Schrödinger’s equation become rapidly more costly for large molecules and large basis sets.
Method
The paper maps electronic Hamiltonians into quantum circuits using molecular integrals, Trotterized propagation, Jordan-Wigner transformation, phase estimation, and adiabatic state preparation.
Results
The construction is explicitly illustrated for H2 as a single time step of a first-order Trotter expansion.
Takeaways & Limitations
The paper provides an explicit scalable-form construction of the H2 unitary propagator for prospective small-scale quantum-computer experiments.
Takeaways & Limitations
Phase-estimation precision requires an exponential increase in uses of U(t0) and corresponding gate count with the number of desired bits.

Abstract

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Over the last century, a large number of physical and mathematical developments paired with rapidly advancing technology have allowed the field of quantum chemistry to advance dramatically. However, the lack of computationally efficient methods for the exact simulation of quantum systems on classical computers presents a limitation of current computational approaches. We report, in detail, how a set of pre-computed molecular integrals can be used to explicitly create a quantum circuit, i.e. a sequence of elementary quantum operations, that, when run on a quantum computer, to obtain the energy of a molecular system with fixed nuclear geometry using the quantum phase estimation algorithm. We extend several known results related to this idea and discuss the adiabatic state preparation procedure for preparing the input states used in the algorithm. With current and near future quantum devices in mind, we provide a complete example using the hydrogen molecule, of how a chemical Hamiltonian can be simulated using a quantum computer.

1. Introduction

Classical simulation becomes rapidly more costly as chemical descriptions improve, motivating quantum simulation approaches. The paper develops a general electronic-Hamiltonian decomposition and demonstrates it with an explicit minimal-basis H2 circuit.

1. Introduction: Classical methods for solving Schrödinger’s equation become rapidly more expensive for large molecules and large basis sets.The computational cost rises as the quality of the description increases, despite growing classical computational power.
1. Introduction: Quantum simulation uses quantum devices to simulate dynamics and can offer exponential advantages for electronic energies and other molecular quantities.The passage lists electronic energies, reaction rates, correlation functions, and molecular properties as target applications.
1. Introduction: Quantum error correction is not considered, although fault-tolerant constructions may require costly redundant qubits for near-term simulation.The paper identifies error effects as important and notes resource concerns for straightforward simulation applications.
1. Introduction: Existing hydrogen-molecule experiments relied on the low-dimensional propagator available for the minimal-basis H2 model.That representation was identified as the key limitation of both experimental algorithms.
1. Introduction: Beyond two dimensions, the decomposition that simplifies higher powers of the unitary propagator is unavailable.For two-level systems, changing α to jα and γ to jγ avoids further decomposition for U^j.
1. Introduction: The paper provides a general decomposition for electronic Hamiltonians and an explicit quantum circuit for one first-order Trotter step of minimal-basis H2.This extends supplementary material associated with earlier work.

2. Overview of the quantum algorithm

The algorithm maps a molecular Hamiltonian into quantum gates that approximate time evolution, then applies phase estimation to recover an eigenvalue. The workflow is illustrated with a hydrogen-molecule construction.

2. Overview of the quantum algorithm: Fourier analysis of an electronic autocorrelation function provides access to eigenenergies and molecular spectra.The measured procedure collapses an input wave function into an eigenstate and returns its autocorrelation frequency.
2. Overview of the quantum algorithm: The simulation workflow decomposes the Hamiltonian into Hermitian terms, constructs a Trotterized propagator, maps it to gates, and estimates an eigenvalue.Figure 1 summarizes these stages, including the requirement for a correctly prepared stationary state.
2. Overview of the quantum algorithm: The Hamiltonian is written as products of Pauli operators using the Jordan-Wigner transformation.This expresses fermionic operators in a form acting on distinguishable qubits.
2. Overview of the quantum algorithm: Each Pauli-operator term is converted into sequential gates that approximate exp(−iHt) through Trotterized time evolution.The decomposition is designed for noncommuting terms by exponentiating them individually over small time steps.
2. Overview of the quantum algorithm: Phase estimation uses the quantum Fourier transform of time-domain propagation to approximate an input eigenstate’s eigenvalue.Eigenstate preparation is discussed separately as part of the simulation procedure.
2. Overview of the quantum algorithm: The complete construction is applied to hydrogen, with the key simulation components depicted in Figure 1.The example is presented as an illustration of the algorithmic steps.

3. The Electronic Hamiltonian

The electronic-structure formulation uses fixed nuclear geometry, second quantization, molecular integrals, and qubit operators to represent the molecular Hamiltonian. These ingredients support its implementation as a quantum circuit.

3. The Electronic Hamiltonian: The Born-Oppenheimer approximation separates nuclear and electronic wave functions, enabling the electronic Schrödinger equation to be solved at fixed nuclear geometry.The separation follows from the mass difference between electrons and nuclei.
3. The Electronic Hamiltonian: Second quantization represents adding and removing electrons with creation and annihilation operators while enforcing fermionic antisymmetry.The N-electron wave function is expanded over configuration state functions of Fock space.
3. The Electronic Hamiltonian: The Hamiltonian’s one- and two-electron matrix elements are computed from a finite basis of molecular functions.Gaussian single-particle basis functions make these integrals efficiently computable.
3. The Electronic Hamiltonian: Spin orbitals combine spatial molecular orbitals with spin functions to form the orbital basis used in the electronic description.The notation defines χp from a spatial function and an electron-spin function.
3. The Electronic Hamiltonian: The one-electron integrals include electronic kinetic energy and electron-nuclear attraction, while two-electron integrals describe electron-electron interaction.The latter involves the 1/r12 interaction.
3. The Electronic Hamiltonian: Qubits encode quantum states using wave functions that permit superposition and entanglement.The circuit model uses elementary gates to reproduce arbitrary unitary transformations to arbitrary precision with a universal gate set.
3. The Electronic Hamiltonian: The Jordan-Wigner transform expresses fermionic operators as Pauli spin operators acting on distinguishable qubits.This addresses the mismatch between experimentally addressable qubits and indistinguishable electrons.

4. Efficient approximations of the unitary propagator by a Trotter decomposition

The paper approximates molecular time evolution by decomposing a Hamiltonian into locally implementable exponentials, then realizes those terms with universal quantum gates. Trotter refinement reduces approximation error but increases circuit complexity.

Circuit construction: The resulting circuits provide sequential gate operations whose combined action approximates the molecular unitary propagator exp(−iHt).These circuit representations form the basis for constructing molecular propagators.
Trotter decomposition: The Trotter decomposition approximates evolution under non-commuting Hamiltonian terms by repeatedly applying exponentials of individual terms over small time steps.The procedure uses t/Δt steps, with a remainder slice when the ratio is not an integer.
Trotter decomposition: As Δt approaches zero, the first-order Trotter error vanishes, while higher-order Suzuki–Trotter formulas can reduce error further.Higher-order formulas have increasingly complicated forms.
Circuit construction: Each second-quantized Hamiltonian term is mapped through the Jordan–Wigner transformation to Pauli operators and exponentiated using a universal gate set.The resulting circuit constructions are summarized for general fermionic Hamiltonians.
Circuit construction: Exponentials of Pauli products are implemented with basis changes, CNOT entangling gates, and Rz rotations.For σz⊗σz, two CNOT gates surround the Rz operation; the construction extends to more qubits.

5. The phase estimation algorithm

Phase estimation encodes a Hamiltonian eigenvalue as a relative phase generated by controlled time evolution, then extracts its binary digits iteratively. The method uses previously measured bits to correct subsequent measurements, while precision requires exponentially more propagator uses.

Phase encoding: Controlled evolution under U(t)=exp(−iHt) converts an eigenenergy into a relative phase, with longer evolution powers exposing successive binary digits.Applying U(2^k t) shifts the phase so bit jk can be read.
Energy recovery: The measured phase is converted back to energy by selecting an evolution time from the energy bounds and applying an integer-aligning energy shift.The relation uses t=2π/(Emax−Emin) and a register-qubit phase gate.
Phase encoding: The phase estimation algorithm estimates an eigenphase φ from the fractional binary digits returned by measurements.The desired output is an L-bit estimate of the phase.
Iterative measurement: The iterative algorithm reads the least significant bit first and applies a counter-rotation determined by previously obtained bits.The correction removes lower-order binary digits before the next Hadamard measurement.
Iterative measurement: After the counter-rotation and Hadamard transform, a measurement yields jk deterministically for a finite binary phase expansion.A non-terminating expansion introduces some randomness from the remainder.
Precision cost: The number of uses of U(t0), and therefore the gate count, scales exponentially with the number of desired phase bits.This limits the cost of increasing measurement precision.
Input-state requirement: For ground-state energies, phase estimation requires an input wavefunction with significant ground-state overlap; otherwise state preparation must improve the overlap.Hartree–Fock provides a straightforward computational-basis input when its overlap is sufficient.

6. Adiabatic state preparation

The paper uses adiabatic evolution to improve an initial Hartree–Fock state when its ground-state overlap is insufficient for phase estimation. The procedure follows a Hamiltonian path toward the molecular Hamiltonian, but its scaling depends on the spectral gap and path.

Preparation procedure: Adiabatic state preparation starts from a Hartree–Fock computational-basis state and slowly applies the molecular Hamiltonian to recover its ground state.Occupied molecular orbitals are represented by qubits in |1⟩ and unoccupied orbitals by |0⟩.
Preparation procedure: The procedure uses a smooth one-parameter Hamiltonian path H(s) for monotonic s∈[0,1].The study restricts attention to the path originally proposed by the authors, while noting that alternatives exist.
Efficiency condition: Efficient preparation requires a total runtime satisfying the adiabatic condition determined by the instantaneous energy spectrum.The condition follows known connections between the adiabatic theorem and complexity theory.
Error behavior: If adiabatic evolution induces higher-energy transitions, the wavefunction error is linear in λ while the expectation-value energy overestimate is quadratic.The cited analysis assumes λ<1.
Scope and limitations: The scalability of adiabatic preparation is unclear because efficient preparation depends on a sufficiently large ground-to-excited-state gap and the chosen path.The paper also notes that other state-preparation strategies can go beyond Hartree–Fock.

7. Simulating the hydrogen molecule

The hydrogen-molecule example instantiates the general construction in a minimal basis, converting molecular integrals into a gate-level propagator. Classical first-order Trotter simulations examine the resulting error–gate trade-off.

Minimal-basis model: The H2 example uses a minimal basis with one spatial function per atom and symmetry-determined bonding and antibonding orbitals.In this simple case, a Hartree–Fock procedure is unnecessary to determine the molecular spatial orbitals.
Minimal-basis model: The one- and two-electron integrals are evaluated for restricted Hartree–Fock orbitals at an internuclear distance of 1.401000 atomic units.This corresponds to 7.414·10^-11 m.
Gate-level Hamiltonian: After Jordan–Wigner transformation, the relevant two-electron interactions reduce to commuting Pauli products that may be implemented in any order.The factor of 1/2 accounts for indistinguishability of the electrons.
Trotter assessment: Decreasing dt reduces approximation error at the expense of more gates, while higher-order decompositions permit more precise simulation with larger time steps.For an appropriate Trotter order, the gate count scales as O(tλ), where λ is the largest simulated eigenvalue.
Gate-level Hamiltonian: For hydrogen, h(1)=−h(2) and h(3)=0, leaving four Pauli terms for the resulting circuit.The terms are σxσxσyσy, σyσyσxσx, σyσxσxσy, and σxσyσyσx.
Trotter assessment: The first-order Trotter circuit is simulated classically to assess propagator error and gate growth as the Trotter number increases.The analysis balances lower Trotter error against the precision needed to detect it.

8. Conclusions

The paper maps full configuration interaction into a quantum algorithm and explicitly constructs a first-order Trotter circuit for H2. Its applicability depends on controlling propagation error and balancing longer evolution against measurement demands.

The paper maps the full configuration interaction method from quantum chemistry into a quantum algorithm and reviews simulated-propagator construction.
The authors explicitly illustrate the construction for H2 using a single time step of a first-order Trotter expansion.
Phase estimation requires longer propagation to determine eigenenergy frequencies more precisely, which increases the number of computational manipulations.
Small-scale experiments implementing the presented H2 circuits on quantum computers are expected to become possible in the near future.

Appendix A. Example program for a quantum computer to calculate electronic energies for the hydrogen molecule

The appendix specifies the gates needed for one H2 Hamiltonian time step and explains how repeated Trotter steps approximate longer evolution. It provides tabulated single- and two-electron gate sequences for the minimal-basis propagator.

A single H2 evolution step alternates non-commuting unitary operators for fractional time intervals, with approximation error vanishing as step length approaches zero.
The propagator applies the full unitary operator when the register qubit is in state |1⟩, using molecular integrals parameterized by nuclear configuration.
The appendix lists gate targets, control qubits, and parameters for single-electron operators and two-electron number-number operators.
Two-electron excitation-excitation operators are implemented through basis changes, controlled-NOT sequences, controlled rotations, and integral-dependent phases.