Even more efficient quantum computations of chemistry through tensor hypercontraction (2011.03494v3)

Abstract: We describe quantum circuits with only $\widetilde{\cal O}(N)$ Toffoli complexity that block encode the spectra of quantum chemistry Hamiltonians in a basis of $N$ arbitrary (e.g., molecular) orbitals. With ${\cal O}(\lambda / \epsilon)$ repetitions of these circuits one can use phase estimation to sample in the molecular eigenbasis, where $\lambda$ is the 1-norm of Hamiltonian coefficients and $\epsilon$ is the target precision. This is the lowest complexity that has been shown for quantum computations of chemistry within an arbitrary basis. Furthermore, up to logarithmic factors, this matches the scaling of the most efficient prior block encodings that can only work with orthogonal basis functions diagonalizing the Coloumb operator (e.g., the plane wave dual basis). Our key insight is to factorize the Hamiltonian using a method known as tensor hypercontraction (THC) and then to transform the Coulomb operator into an isospectral diagonal form with a non-orthogonal basis defined by the THC factors. We then use qubitization to simulate the non-orthogonal THC Hamiltonian, in a fashion that avoids most complications of the non-orthogonal basis. We also reanalyze and reduce the cost of several of the best prior algorithms for these simulations in order to facilitate a clear comparison to the present work. In addition to having lower asymptotic scaling spacetime volume, compilation of our algorithm for challenging finite-sized molecules such as FeMoCo reveals that our method requires the least fault-tolerant resources of any known approach. By laying out and optimizing the surface code resources required of our approach we show that FeMoCo can be simulated using about four million physical qubits and under four days of runtime, assuming $1\,\mu$s cycle times and physical gate error rates no worse than $0.1\%$.

• The paper introduces tensor hypercontraction to encode molecular Hamiltonians for efficient quantum simulations.
• It reduces the Toffoli complexity from N^3.9 to N^3.1, significantly lowering computational resources for electronic structure calculations.
• The method offers practical benefits for fault-tolerant quantum platforms, improving qubit efficiency and simulation runtimes.

Efficient Quantum Computation in Quantum Chemistry Through Tensor Hypercontraction

The paper under discussion introduces a novel approach to quantum computations in quantum chemistry by leveraging tensor hypercontraction (THC). This method offers an efficient framework for simulating quantum chemistry Hamiltonians by employing a sophisticated combination of tensor algebra and quantum algorithms, specifically focusing on THC for reducing computational complexity in quantum simulations.

Key Contributions

1. Quantum Simulations Using Tensor Hypercontraction: The research highlights an advanced application of tensor hypercontraction to encode the molecular Hamiltonian, facilitating efficient quantum computation of electronic structures. This approach derives a non-orthogonal basis from auxiliary tensors that diagonalize the Coulomb operator, allowing for efficient simulation through qubitization while maintaining the original basis set's size.
2. Reduction in Computational Complexity: By reducing the Toffoli complexity - a measure crucial for quantum computations - to $\widetilde{\cal O}(N \lambda / \epsilon)$, the proposed method becomes more favorable than previous methods. The innovative use of a non-orthogonal basis with effective transformations between tensor components significantly reduces the required computational resources without increasing qubit overhead.
3. Comparative Analysis with Prior Work: The methodology rigorously compares to previous strategies, including the "sparse" and double factorization techniques. The proposed THC approach showcases superior performance both in asymptotic scaling and resource requirements for benchmark systems such as the FeMoCo cluster. The reduction in connectivity and computational gates demonstrates a clear advantage in simulated systems' runtime and error-correction requirements.

Numerical and Theoretical Insights

• Empirical Scaling: Through extensive analysis on hydrogen chains and FeMoCo models, the work confirms that the tensor hypercontraction approach scales better in thermodynamic and continuum limits compared to its predecessors. The reduction from $N^{3.9}$ to $N^{3.1}$ in Toffoli complexity showcases a substantial improvement in algorithmic efficiency.
• Improvements in Physical Implementation: The paper also examines practical aspects of physical qubit implementations, relevant to fault-tolerant quantum computing platforms like the surface code. Innovative layout and optimization strategies allow the proposed method to require fewer physical qubits and a shorter runtime, even under assumptions of realistic error rates.

Future Implications and Speculative Directions

1. Towards General-Purpose Quantum Chemistry Algorithms: The paper sets a relevant milestone in reducing quantum algorithm complexity for arbitrary basis quantum chemistry. It opens pathways to further explore and optimize ordinal representation to maintain chemical accuracy while minimizing quantum resource usage.
2. Broadening Computational Chemical Horizons: By demonstrating efficient phase estimation and eigenstate preparation in complex molecular systems, the research extends the practical horizons of quantum simulation in real-world chemical problems, significantly impacting fields like material science and pharma.
3. Adoption to Other Quantum Algorithms: While primarily applied to chemistry's electronic structure problem, the techniques in qubitization and tensor hypercontraction could inspire adaptation into other domains requiring accurate Hamiltonian simulation, including condensed matter and quantum field theories.

In conclusion, the integration of tensor hypercontraction within quantum computing paradigms as presented in this paper marks a significant advance towards scalable and resource-efficient quantum chemistry simulations, with the potential to transcend traditional computational boundaries and enable practical quantum advantage.