Periodic Reporting for period 1 - COMFTQUA (Enabling efficient computation on fault tolerant quantum computers)
Reporting period: 2023-07-01 to 2024-06-30
Fault-tolerant quantum computing (FTQC) systems hold the potential to deliver breakthroughs across major areas of modern science. Due to often insufficient classical computing power for performing adequately accurate simulations of molecules and materials, the prospective polynomial-to- exponential speedup offered by FTQC is the most dynamically pursued goal in the quantum-computing community. The societal and industrial impact of advancements in molecular simulations with quantum computers can can be profound.
Our project aims to develop quantum algorithms, that is instructions for a FT quantum computer, that calculate important properties of molecules and materials, to support the design of efficient lithium-ion batteries, catalysts, and pharmaceutical drugs. A key metric that determines the usability of a prospective FT quantum algorithm is the number of operations (quantum gates) that an algorithm needs to perform.
We thus formulated the following objectives:
1. develop novel molecular simulation techniques and their hardware-agnostic FTQC implementations
2. reduce the quantum computing resources (the number of qubits and quantum gates) necessary to perform such simulations on FTQC systems.
In the course of the project so far, we have proposed four novel quantum algorithms that require fewer quantum gates than existing techniques, leading to shorter quantum circuits and faster, more reliable quantum computer simulations.
To illustrate the impact created by an efficient FTQC implementation of an algorithm simulating molecular properties we bring forward the problem of nitrogen conversion into fertilisers. The process of nitrogen fixation consumes about 4% of the global energy production. The FeMo cofactor (“FeMoCo”) found in the nitrogenase enzyme, plays a vital role in biological nitrogen fixation. Performing electronic energy calculations for FeMoCo at an appropriate accuracy presents a computational challenge beyond the capabilities of today’s high-performance computers. We therefore studied the FeMoCo molecule, for which we showed 40% reduction in the quantum computing resources for the energy levels simulations. With algorithmic advancements of this type, we believe it will be possible to design better catalyst molecules, which eventually resolve the problem of fertiliser production.
Our project aims to develop quantum algorithms, that is instructions for a FT quantum computer, that calculate important properties of molecules and materials, to support the design of efficient lithium-ion batteries, catalysts, and pharmaceutical drugs. A key metric that determines the usability of a prospective FT quantum algorithm is the number of operations (quantum gates) that an algorithm needs to perform.
We thus formulated the following objectives:
1. develop novel molecular simulation techniques and their hardware-agnostic FTQC implementations
2. reduce the quantum computing resources (the number of qubits and quantum gates) necessary to perform such simulations on FTQC systems.
In the course of the project so far, we have proposed four novel quantum algorithms that require fewer quantum gates than existing techniques, leading to shorter quantum circuits and faster, more reliable quantum computer simulations.
To illustrate the impact created by an efficient FTQC implementation of an algorithm simulating molecular properties we bring forward the problem of nitrogen conversion into fertilisers. The process of nitrogen fixation consumes about 4% of the global energy production. The FeMo cofactor (“FeMoCo”) found in the nitrogenase enzyme, plays a vital role in biological nitrogen fixation. Performing electronic energy calculations for FeMoCo at an appropriate accuracy presents a computational challenge beyond the capabilities of today’s high-performance computers. We therefore studied the FeMoCo molecule, for which we showed 40% reduction in the quantum computing resources for the energy levels simulations. With algorithmic advancements of this type, we believe it will be possible to design better catalyst molecules, which eventually resolve the problem of fertiliser production.
We developed the following techniques:
1. The “Spin-SWAP-network” algorithm for simulations of the relativistic Pauli-Breit Hamiltonian
We designed a quantum algorithm to simulate the energy levels of molecules relevant to lithium-ion batteries, photomaterials, and catalytic reactions using the relativistic Pauli-Breit Hamiltonian. Representing the Hamiltonian with the spin-unitary group approach made it suitable for a quantum computer implementation. Using quantum-phase estimation and block encoding, we decoupled electron spin-index controls from orbital-index controls in the SELECT circuit, reducing the circuit’s T-gate count by 2-3 times compared to an unoptimized implementation [1].
2. Exploiting symmetries and tensor-network factorisations (STNF) in the electronic ground state calculations
Given the critical importance of accurately calculating the energy levels of structures like the FeMoCo molecule, we performed quantum resource estimation for several classes of industrially important molecules. To reduce the number of quantum T-gates in electronic ground state calculations, we combined the symmetry shift method and the double-factorisation method [1,5]. Our results show an average 40% reduction in quantum T-gate count compared to the standard methodology of Reiher et al.[4] Encouraged by these results, we extended our calculations to other molecules, including Ruthenium-based catalysts studied by Burg et al., which convert CO2 into methanol. The reduction in quantum resources was around 37% for all catalysts considered. We further developed symmetry-based techniques and proposed a new method based on molecular symmetry groups to reduce quantum computing resources in the quantum phase estimation algorithm. Research is ongoing.
3. Quantum resource estimator (QRE)
We developed a computer code that accurately counts the number of T-gates, the T-depth and the number of qubits in the Quantum Phase Estimation (QPE) algorithm. Our QRE code has been extensively used in points 1-2), serving as a verification toolkit for our rese ideas. Throughout the production process of the QRE code, our team gained an in-depth understanding of quantum algorithms presented in the state-of-the-art literature [1-2]. This collective team knowledge presents an asset that will be utilised as the project continues.
The QRE is pivotal in our relations with clients. It allows our business partners to assess the FT quantum resources necessary to perform calculations of molecular energies to an arbitrary accuracy and precision. The QRE’s inputs are parameters of the simulation (e.g. orbitals, accuracy, precision and tensor-network cut-offs). The QRE returns the exact Toffoli gate count (T-gate count) needed for the computation. Our code is being continuously developed, with more functionalities and a range of quantum algorithms added in order to cover a wider range of applications.
4. Quantum computing circuits implementing the discrete-variable representation (DVR) transformation
We developed a novel quantum circuit implementing the so-called Discrete Variable Representation (DVR) transformation. DVRs are widely used in computational physics and engineering including space engineering, exoplanet spectroscopy and attosecond physics.
Our contribution includes the design and optimisation of a general quantum DVR oracle together with a showcase of a specific implementation based on the Gauss-Hermite quadratures, which is commonly used in bosonic systems simulations.
Our implementation uses fewer FTQC resources than offered by generic state-of-the-art methods of unitary synthesis [3] (quantum volume of O(N) vs. O(N^2).
[1] V von Burg et al., Phys. Rev. Research 3, 033055 (2021).
[2] I. Kim, Phys. Rev. Research 4, 023019 (2022).
[3] G. Low et al., arXiv:1812.00954 (2018).
[4] M. Reiher, PNAS 114 (29) 7555-7560 (2017).
[5] I. Loaiza et al.,arXiv:2304.13772 (2022).
1. The “Spin-SWAP-network” algorithm for simulations of the relativistic Pauli-Breit Hamiltonian
We designed a quantum algorithm to simulate the energy levels of molecules relevant to lithium-ion batteries, photomaterials, and catalytic reactions using the relativistic Pauli-Breit Hamiltonian. Representing the Hamiltonian with the spin-unitary group approach made it suitable for a quantum computer implementation. Using quantum-phase estimation and block encoding, we decoupled electron spin-index controls from orbital-index controls in the SELECT circuit, reducing the circuit’s T-gate count by 2-3 times compared to an unoptimized implementation [1].
2. Exploiting symmetries and tensor-network factorisations (STNF) in the electronic ground state calculations
Given the critical importance of accurately calculating the energy levels of structures like the FeMoCo molecule, we performed quantum resource estimation for several classes of industrially important molecules. To reduce the number of quantum T-gates in electronic ground state calculations, we combined the symmetry shift method and the double-factorisation method [1,5]. Our results show an average 40% reduction in quantum T-gate count compared to the standard methodology of Reiher et al.[4] Encouraged by these results, we extended our calculations to other molecules, including Ruthenium-based catalysts studied by Burg et al., which convert CO2 into methanol. The reduction in quantum resources was around 37% for all catalysts considered. We further developed symmetry-based techniques and proposed a new method based on molecular symmetry groups to reduce quantum computing resources in the quantum phase estimation algorithm. Research is ongoing.
3. Quantum resource estimator (QRE)
We developed a computer code that accurately counts the number of T-gates, the T-depth and the number of qubits in the Quantum Phase Estimation (QPE) algorithm. Our QRE code has been extensively used in points 1-2), serving as a verification toolkit for our rese ideas. Throughout the production process of the QRE code, our team gained an in-depth understanding of quantum algorithms presented in the state-of-the-art literature [1-2]. This collective team knowledge presents an asset that will be utilised as the project continues.
The QRE is pivotal in our relations with clients. It allows our business partners to assess the FT quantum resources necessary to perform calculations of molecular energies to an arbitrary accuracy and precision. The QRE’s inputs are parameters of the simulation (e.g. orbitals, accuracy, precision and tensor-network cut-offs). The QRE returns the exact Toffoli gate count (T-gate count) needed for the computation. Our code is being continuously developed, with more functionalities and a range of quantum algorithms added in order to cover a wider range of applications.
4. Quantum computing circuits implementing the discrete-variable representation (DVR) transformation
We developed a novel quantum circuit implementing the so-called Discrete Variable Representation (DVR) transformation. DVRs are widely used in computational physics and engineering including space engineering, exoplanet spectroscopy and attosecond physics.
Our contribution includes the design and optimisation of a general quantum DVR oracle together with a showcase of a specific implementation based on the Gauss-Hermite quadratures, which is commonly used in bosonic systems simulations.
Our implementation uses fewer FTQC resources than offered by generic state-of-the-art methods of unitary synthesis [3] (quantum volume of O(N) vs. O(N^2).
[1] V von Burg et al., Phys. Rev. Research 3, 033055 (2021).
[2] I. Kim, Phys. Rev. Research 4, 023019 (2022).
[3] G. Low et al., arXiv:1812.00954 (2018).
[4] M. Reiher, PNAS 114 (29) 7555-7560 (2017).
[5] I. Loaiza et al.,arXiv:2304.13772 (2022).
Our results advance the state-of-the-art in two areas: molecular simulation methodologies on FTQC and the quantum resources required.
In molecular simulation methodologies, our contributions include:
- Quantum circuits for Pauli-Breit Hamiltonian simulation
- Molecular symmetry shift methods for Hamiltonian norm reduction
- Novel circuits for discrete-variable representation transformation
In quantum resource reduction, our advancements include:
- Spin-SWAP network for Pauli-Breit Hamiltonian simulation, reducing T-gate count by 2-3 times
- Symmetry-based STNF method, reducing Hamiltonian norm by about 40%
- Quantum arithmetic and recursive algorithm design for discrete-variable representation transformation, achieving O(betaN) T-gate count scaling compared to O(betaN^2) for standard techniques.
In molecular simulation methodologies, our contributions include:
- Quantum circuits for Pauli-Breit Hamiltonian simulation
- Molecular symmetry shift methods for Hamiltonian norm reduction
- Novel circuits for discrete-variable representation transformation
In quantum resource reduction, our advancements include:
- Spin-SWAP network for Pauli-Breit Hamiltonian simulation, reducing T-gate count by 2-3 times
- Symmetry-based STNF method, reducing Hamiltonian norm by about 40%
- Quantum arithmetic and recursive algorithm design for discrete-variable representation transformation, achieving O(betaN) T-gate count scaling compared to O(betaN^2) for standard techniques.