[Paper Review] Majorana Tensor Decomposition: A unifying framework for decompositions of fermionic Hamiltonians to Linear Combination of Unitaries

Ignacio Loaiza, Artur F. Izmaylov+1 more|arXiv (Cornell University)|Jul 9, 2024

Advanced NMR Techniques and Applications4 citations

TL;DR

This paper introduces the Majorana Tensor Decomposition (MTD), a unified framework for decomposing fermionic Hamiltonians into Linear Combinations of Unitaries (LCU) using low-rank tensor factorizations. By expressing the Hamiltonian in terms of Majorana operators and leveraging tensor decompositions like Tucker and Matrix Product States, MTD enables more efficient LCU implementations with reduced T-gate counts and ancilla qubit requirements, significantly lowering quantum circuit costs for quantum chemistry simulations.

ABSTRACT

Linear combination of unitaries (LCU) decompositions have appeared as one of the main tools for encoding operators on quantum computers, allowing efficient implementations of arbitrary operators. In particular, LCU approaches present a way of encoding information from the electronic structure Hamiltonian into a quantum circuit. Over the past years, many different decomposition techniques have appeared for the electronic structure Hamiltonian. Here we present the Majorana Tensor Decomposition (MTD), a framework that unifies existing LCUs and offers novel decomposition methods by using popular low-rank tensor factorizations.

Motivation & Objective

• To unify existing LCU decomposition methods for fermionic Hamiltonians into a single, systematic framework.
• To reduce the resource cost of LCU-based quantum algorithms by minimizing the 1-norm λ and optimizing circuit depth and T-gate count.
• To extend LCU encoding to non-fermionic Hamiltonians, such as vibrational systems, via Majorana operator polynomials.
• To provide a scalable, tensor-factorization-based approach that enables efficient implementation of Hamiltonian oracles in quantum algorithms.
• To benchmark and compare MTD variants (e.g., MTD-L⁴, L⁴-MPS) against standard LCU methods in terms of T-gate count and qubit usage.

Proposed method

• The framework expresses the electronic structure Hamiltonian as a sum of Majorana operators, enabling a unified representation of LCU decompositions.
• It applies low-rank tensor factorizations—such as Tucker and Matrix Product State (MPS)—to the two-electron integral tensor to reduce the number of unitaries in the LCU decomposition.
• The method constructs a block-encoding oracle via a controlled-SWAP and controlled-rotation scheme, using QROMs and multiplexed Givens rotations to load coefficients.
• The SELECT circuit is implemented using fault-tolerant Toffoli and T-gate constructions, with ancilla qubits optimized via shared resource allocation.
• The 1-norm λ is minimized by exploiting tensor structure, directly reducing the cost of phase estimation via qubitization.
• The approach supports both non-optimized and optimized LCU implementations, including MTD-L⁴ and L⁴-MPS variants, with explicit T-gate and qubit cost formulas.

Experimental results

Research questions

• RQ1 Can a unified framework be developed to express diverse LCU decompositions of fermionic Hamiltonians using tensor factorizations?
• RQ2 How do tensor-based LCU decompositions like MTD-L⁴ and L⁴-MPS compare in T-gate and qubit cost to standard LCU methods?
• RQ3 To what extent can the 1-norm λ of the LCU decomposition be reduced through Majorana operator polynomials and tensor structure?
• RQ4 Can the MTD framework be extended to non-fermionic Hamiltonians, such as vibrational systems, via Majorana operator representations?
• RQ5 What is the trade-off between T-gate count and ancilla qubit usage in different MTD variants?

Key findings

• The MTD-L⁴ decomposition achieves a T-gate count of N(112β - 196) + 8W - 4, with 4 + 2N + b_W + 4β non-reusable qubits and b_W reusable qubits, showing a significant reduction in T-gate usage compared to standard LCU.
• The L⁴-MPS variant exhibits a T-gate cost of N(4α₂ + 112β - 192) + α₂(8α₁ + 8α₃) + 8α₁ + 4α₃ - 24, demonstrating scalability through hierarchical tensor structure.
• The MTD framework reduces the 1-norm λ by exploiting low-rank structure in two-electron integrals, with convergence achieved at a 2-norm error < 1×10⁻⁶ in the STO-3G basis.
• The method enables efficient implementation of the Hamiltonian oracle with only 4 Hadamard gates and 8L R_Z rotations, minimizing fault-tolerant overhead.
• The framework supports shared ancilla qubits across multiple PREPARE circuits, reducing total non-reusable qubit count by up to 50% in multi-component implementations.
• The use of controlled QROMs and multiplexed Givens rotations allows for scalable loading of LCU coefficients with minimal T-gate overhead.