[Paper Review] General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory

Zohreh Davoudi, Alexander F. Shaw+1 more|arXiv (Cornell University)|Dec 28, 2022

Physics of Superconductivity and Magnetism5 citations

TL;DR

This paper presents a general quantum algorithm for simulating Hamiltonians with correlated changes in multiple quantum numbers using singular-value decomposition to diagonalize interaction terms. Applied to a 1+1D SU(2) lattice gauge theory with staggered fermions, the method achieves significant resource reductions—up to 380-fold lower CNOT cost—by using a loop-string-hadron formulation that preserves gauge symmetry without costly controlled operations, outperforming standard Schwinger-boson and angular-momentum formulations.

ABSTRACT

With a focus on universal quantum computing for quantum simulation, and through the example of lattice gauge theories, we introduce rather general quantum algorithms that can efficiently simulate certain classes of interactions consisting of correlated changes in multiple (bosonic and fermionic) quantum numbers with non-trivial functional coefficients. In particular, we analyze diagonalization of Hamiltonian terms using a singular-value decomposition technique, and discuss how the achieved diagonal unitaries in the digitized time-evolution operator can be implemented. The lattice gauge theory studied is the SU(2) gauge theory in 1+1 dimensions coupled to one flavor of staggered fermions, for which a complete quantum-resource analysis within different computational models is presented. The algorithms are shown to be applicable to higher-dimensional theories as well as to other Abelian and non-Abelian gauge theories. The example chosen further demonstrates the importance of adopting efficient theoretical formulations: it is shown that an explicitly gauge-invariant formulation using loop, string, and hadron degrees of freedom simplifies the algorithms and lowers the cost compared with the standard formulations based on angular-momentum as well as the Schwinger-boson degrees of freedom. The loop-string-hadron formulation further retains the non-Abelian gauge symmetry despite the inexactness of the digitized simulation, without the need for costly controlled operations. Such theoretical and algorithmic considerations are likely to be essential in quantumly simulating other complex theories of relevance to nature.

Motivation & Objective

• To develop general quantum algorithms for simulating Hamiltonians with complex, correlated interactions in multiple quantum numbers.
• To reduce quantum resource costs in digital quantum simulation of gauge theories by optimizing theoretical formulations.
• To demonstrate that explicitly gauge-invariant formulations like loop-string-hadron can simplify simulation and avoid costly controlled operations.
• To provide a complete quantum-resource analysis across near-term and fault-tolerant models for a non-Abelian gauge theory.
• To show that efficient theoretical formulations can drastically reduce gate counts while preserving gauge symmetry in digitized time evolution.

Proposed method

• Uses singular-value decomposition (SVD) to diagonalize Hamiltonian terms with non-trivial functional coefficients.
• Applies digitized time-evolution via product formulas, decomposing the evolution operator into diagonal unitaries.
• Implements diagonal unitaries using phase estimation (far-term) or variational circuits (near-term) with qubit-efficient encoding.
• Adopts a loop-string-hadron formulation of the SU(2) lattice gauge theory that explicitly maintains gauge invariance.
• Performs error analysis using second-order Trotter error bounds derived from commutator norms and spectral norms.
• Compares resource costs across formulations (Schwinger-boson vs. loop-string-hadron) using gate-count tables for CNOT and T-gates.

Experimental results

Research questions

• RQ1 Can SVD-based diagonalization be generalized to efficiently simulate Hamiltonians with correlated changes in multiple quantum numbers?
• RQ2 How do different theoretical formulations (Schwinger-boson, angular momentum, loop-string-hadron) affect quantum resource costs in gauge theory simulations?
• RQ3 Can a gauge-invariant formulation like loop-string-hadron avoid the need for costly controlled operations while preserving non-Abelian symmetry in digitized simulations?
• RQ4 What is the quantitative impact of formulation choice on CNOT and T-gate counts in near-term and fault-tolerant quantum simulations?
• RQ5 How do error bounds scale with system size, coupling strength, and electric field truncation in digitized time evolution?

Key findings

• The loop-string-hadron formulation reduces CNOT gate cost by up to 380 times compared to the Schwinger-boson formulation at x = 1 and η = 4.
• The loop-string-hadron formulation achieves a 24.5-fold reduction in T-gate cost at x = 1, with a 21- to 24-fold reduction across x = 1 to x = 10.
• The LSH formulation maintains non-Abelian gauge symmetry without controlled operations, unlike standard formulations requiring such gates.
• The second-order Trotter error bound for the LSH formulation scales as ρLSH(x, Λ, µ) ≈ 47√2x³/3 + 2Λx² + ..., showing improved scaling over Schwinger-boson counterparts.
• The loop-string-hadron formulation reduces the number of non-zero commutator terms and their multiplicities, contributing to lower error and gate counts.
• The algorithm is general and applicable to higher-dimensional and other Abelian/non-Abelian gauge theories beyond the SU(2) 1+1D example.