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Quantum chemistry and quantum computers -- Testing the Bravyi-Kitaev mapping and Trotter order optimisations

Andrew Tranter

Published Jun 1, 2018• 2 citations
Publisher

Abstract

The quantum simulation of quantum chemistry is often cited as one of the most important potential uses for emerging quantum technology. Quantum computing methods are expected to be able to perform full configuration interaction level quantum chemistry calculations with tractable computational resource requirements. However, early quantum computers are likely to be constrained in both the number of qubits available, and the number of gates that can be performed. Implementing chemical algorithms on such quantum devices will require algorithmic improvements. In this thesis, two avenues for optimisation of the canonical method for the simulation of quantum chemistry on quantum computers are discussed. The Bravyi-Kitaev transformation is an alternative to the Jordan-Wigner transformation, the canonical method of mapping electronic states and operators to states and operators of qubits. The resource implications of this transformation scheme are assessed using a variety of chemical examples. Known techniques for optimising quantum circuits for quantum chemistry are applied, and the implications for the use of the Bravyi-Kitaev transformation are considered. Trotterization – the use of Trotter-Suzuki formulae to approximate the evolution operator for the molecular Hamiltonian – is discussed. In particular, ordering schemes designed to minimise the Trotter error are described. Finally, the importance of Trotter ordering when using the Bravyi-Kitaev transformation is considered. The performance of either transformation scheme is found to be highly dependent on the Trotter ordering and level of optimisation chosen. As such, the relative merits of either transformation scheme are found to be not as simple as previously thought, although in most cases the Bravyi-Kitaev transformation is found to be advantageous. The relative merits of both mapping and ordering schemes are considered in the context of performing quantum simulation of quantum chemistry on real quantum devices. Particular focus is paid to molecular systems which are intractable to classical computational simulation. The results emphasise the need for thorough analysis of optimisation schemes, such that quantum chemistry may indeed be a principal use of early quantum computation.

DOI
10.25560/66361

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The quantum simulation of quantum chemistry is often cited as one of the most important potential uses for emerging quantum technology. Quantum computing methods are expected to be able to perform full configuration interaction level quantum chemistry calculations with tractable computational resource requirements. However, early quantum computers are likely to be constrained in both the number of qubits available, and the number of gates that can be performed. Implementing chemical algorithms on such quantum devices will require algorithmic improvements. In this thesis, two avenues for optimisation of the canonical method for the simulation of quantum chemistry on quantum computers are discussed. The Bravyi-Kitaev transformation is an alternative to the Jordan-Wigner transformation, the canonical method of mapping electronic states and operators to states and operators of qubits. The resource implications of this transformation scheme are assessed using a variety of chemical examples. Known techniques for optimising quantum circuits for quantum chemistry are applied, and the implications for the use of the Bravyi-Kitaev transformation are considered. Trotterization – the use of Trotter-Suzuki formulae to approximate the evolution operator for the molecular Hamiltonian – is discussed. In particular, ordering schemes designed to minimise the Trotter error are described. Finally, the importance of Trotter ordering when using the Bravyi-Kitaev transformation is considered. The performance of either transformation scheme is found to be highly dependent on the Trotter ordering and level of optimisation chosen. As such, the relative merits of either transformation scheme are found to be not as simple as previously thought, although in most cases the Bravyi-Kitaev transformation is found to be advantageous. The relative merits of both mapping and ordering schemes are considered in the context of performing quantum simulation of quantum chemistry on real quantum devices. Particular focus is paid to molecular systems which are intractable to classical computational simulation. The results emphasise the need for thorough analysis of optimisation schemes, such that quantum chemistry may indeed be a principal use of early quantum computation.
The ability to perform classically intractable electronic structure calculations is often cited as one of the principal applications of quantum computing. A great deal of theoretical algorithmic development has been performed in support of this goal. Most techniques require a scheme for mapping electronic states and operations to states of and operations upon qubits. The two most commonly used techniques for this are the Jordan-Wigner transformation and the Bravyi-Kitaev transformation. However, comparisons of these schemes have previously been limited to individual small molecules. In this paper, we discuss resource implications for the use of the Bravyi-Kitaev mapping scheme, specifically with regard to the number of quantum gates required for simulation. We consider both small systems, which may be simulatable on near-future quantum devices, and systems sufficiently large for classical simulation to be intractable. We use 86 molecular systems to demonstrate that the use of the Bravyi-Kitaev transformation is typically at least approximately as efficient as the canonical Jordan-Wigner transformation and results in substantially reduced gate count estimates when performing limited circuit optimizations.
Quantum simulation is an important application of future quantum computers with applications in quantum chemistry, condensed matter, and beyond. Quantum simulation of fermionic systems presents a specific challenge. The Jordan-Wigner transformation allows for representation of a fermionic operator by O(n) qubit operations. Here, we develop an alternative method of simulating fermions with qubits, first proposed by Bravyi and Kitaev [Ann. Phys. 298, 210 (2002); e-print arXiv:quant-ph/0003137v2], that reduces the simulation cost to O(log n) qubit operations for one fermionic operation. We apply this new Bravyi-Kitaev transformation to the task of simulating quantum chemical Hamiltonians, and give a detailed example for the simplest possible case of molecular hydrogen in a minimal basis. We show that the quantum circuit for simulating a single Trotter time step of the Bravyi-Kitaev derived Hamiltonian for H(2) requires fewer gate applications than the equivalent circuit derived from the Jordan-Wigner transformation. Since the scaling of the Bravyi-Kitaev method is asymptotically better than the Jordan-Wigner method, this result for molecular hydrogen in a minimal basis demonstrates the superior efficiency of the Bravyi-Kitaev method for all quantum computations of electronic structure.
Present quantum computers often work with distinguishable qubits as their computational units. In order to simulate indistinguishable fermionic particles, it is first required to map the fermionic state to the state of the qubits. The Bravyi-Kitaev Superfast (BKSF) algorithm can be used to accomplish this mapping. The BKSF mapping has connections to quantum error correction and opens the door to new ways of understanding fermionic simulation in a topological context. Here, we present the first detailed exposition of the BKSF algorithm for molecular simulation. We provide the BKSF transformed qubit operators and report on our implementation of the BKSF fermion-to-qubits transform in OpenFermion. In this initial study of a hydrogen molecule we have compared BKSF, Jordan-Wigner, and Bravyi-Kitaev transforms under the Trotter approximation. The gate count to implement BKSF is lower than Jordan-Wigner but higher than Bravyi-Kitaev. We considered different orderings of the exponentiated terms and found lower Trotter errors than the previously reported for Jordan-Wigner and Bravyi-Kitaev algorithms. These results open the door to the further study of the BKSF algorithm for quantum simulation.
We present several improvements to the standard Trotter-Suzuki based algorithms used in the simulation of quantum chemistry on a quantum computer. First, we modify how Jordan-Wigner transformations are implemented to reduce their cost from linear or logarithmic in the number of orbitals to a constant. Our modification does not require additional ancilla qubits. Then, we demonstrate how many operations can be parallelized, leading to a further linear decrease in the parallel depth of the circuit, at the cost of a small constant factor increase in number of qubits required. Thirdly, we modify the term order in the Trotter-Suzuki decomposition, significantly reducing the error at given Trotter-Suzuki timestep. A final improvement modifies the Hamiltonian to reduce errors introduced by the non-zero Trotter-Suzuki timestep. All of these techniques are validated using numerical simulation and detailed gate counts are given for realistic molecules.
We present several improvements to the standard Trotter-Suzuki based algorithms used in the simulation of quantum chemistry on a quantum computer. First, we modify how Jordan-Wigner transformations are implemented to reduce their cost from linear or logarithmic in the number of orbitals to a constant. Our modification does not require additional ancilla qubits. Then, we demonstrate how many operations can be parallelized, leading to a further linear decrease in the parallel depth of the circuit, at the cost of a small constant factor increase in number of qubits required. Thirdly, we modify the term order in the Trotter-Suzuki decomposition, significantly reducing the error at given Trotter-Suzuki timestep. A final improvement modifies the Hamiltonian to reduce errors introduced by the non-zero Trotter-Suzuki timestep. All of these techniques are validated using numerical simulation and detailed gate counts are given for realistic molecules.
Although the simulation of quantum chemistry is one of the most anticipated applications of quantum computing, the scaling of known upper bounds on the complexity of these algorithms is daunting. Prior work has bounded errors due to discretization of the time evolution (known as ``Trotterization'') in terms of the norm of the error operator and analyzed scaling with respect to the number of spin orbitals. However, we find that these error bounds can be loose by up to 16 orders of magnitude for some molecules. Furthermore, numerical results for small systems fail to reveal any clear correlation between ground-state error and number of spin orbitals. We instead argue that chemical properties, such as the maximum nuclear charge in a molecule and the filling fraction of orbitals, can be decisive for determining the cost of a quantum simulation. Our analysis motivates several strategies to use classical processing to further reduce the required Trotter step size and estimate the necessary number of steps, without requiring additional quantum resources. Finally, we demonstrate improved methods for state preparation techniques which are asymptotically superior to proposals in the simulation literature.