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arXiv:2209.07918 · SciPost Physics Core

Recipes for the digital quantum simulation of lattice spin systems

Guido Burkard

Published Mar 10, 2025• 4 citations
arXivHTMLPDF

Abstract

We describe methods to construct digital quantum simulation algorithms for quantum spin systems on a regular lattice with local interactions. In addition to tools such as the Trotter-Suzuki expansion and graph coloring, we also discuss the efficiency gained by parallel execution of an extensive number of commuting terms. We provide resource estimates and quantum circuit elements for the most important cases and classes of spin systems. As resource estimates we indicate the total number of gates N N and simulation time T T , expressed in terms of the number n n of spin 1/2 lattice sites (qubits), target accuracy \epsilon ϵ , and simulated time t t . We provide circuit constructions that realize the simulation time T^{(1)}\propto nt^2/\epsilon T ( 1 ) ∝ n t 2 / ϵ and T^{(2q)}\propto t^{1+\eta}n^\eta/\epsilon^\eta T ( 2 q ) ∝ t 1 + η n η / ϵ η for arbitrarily small \eta = 1/2q η = 1 / 2 q for the first-order and higher-order Trotter expansions. We also discuss the potential impact of scaled gates, which have not yet been fully explored.

DOI
10.21468/scipostphyscore.8.1.030

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We describe methods to construct digital quantum simulation algorithms for quantum spin systems on a regular lattice with local interactions. In addition to tools such as the Trotter-Suzuki expansion and graph coloring, we also discuss the efficiency gained by parallel execution of an extensive number of commuting terms. We provide resource estimates and quantum circuit elements for the most important cases and classes of spin systems. As resource estimates we indicate the total number of gates N N and simulation time T T , expressed in terms of the number n n of spin 1/2 lattice sites (qubits), target accuracy \epsilon ϵ , and simulated time t t . We provide circuit constructions that realize the simulation time T^{(1)}\propto nt^2/\epsilon T ( 1 ) ∝ n t 2 / ϵ and T^{(2q)}\propto t^{1+\eta}n^\eta/\epsilon^\eta T ( 2 q ) ∝ t 1 + η n η / ϵ η for arbitrarily small \eta = 1/2q η = 1 / 2 q for the first-order and higher-order Trotter expansions. We also discuss the potential impact of scaled gates, which have not yet been fully explored.
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.
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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.
The Schwinger model (quantum electrodynamics in 1+1 dimensions) is a testbed for the study of quantum gauge field theories. We give scalable, explicit digital quantum algorithms to simulate the lattice Schwinger model in both NISQ and fault-tolerant settings. In particular, we perform a tight analysis of low-order Trotter formula simulations of the Schwinger model, using recently derived commutator bounds, and give upper bounds on the resources needed for simulations in both scenarios. In lattice units, we find a Schwinger model on N / 2 physical sites with coupling constant x − 1 / 2 and electric field cutoff x − 1 / 2 Λ can be simulated on a quantum computer for time 2 x T using a number of T -gates or CNOTs in O ~ ( N 3 / 2 T 3 / 2 x Λ ) for fixed operator error. This scaling with the truncation Λ is better than that expected from algorithms such as qubitization or QDRIFT. Furthermore, we give scalable measurement schemes and algorithms to estimate observables which we cost in both the NISQ and fault-tolerant settings by assuming a simple target observable–the mean pair density. Finally, we bound the root-mean-square error in estimating this observable via simulation as a function of the diamond distance between the ideal and actual CNOT channels. This work provides a rigorous analysis of simulating the Schwinger model, while also providing benchmarks against which subsequent simulation algorithms can be tested.
We report a first-of-its-kind analysis on post-Trotter simulation of U(1), SU(2), and SU(3) lattice gauge theories including fermions in arbitrary spatial dimension. We provide explicit circuit constructions as well as T -gate counts and logical qubit counts for Hamiltonian simulation. We find a reduction of up to 25 orders of magnitude in space-time volume over Trotter methods for simulations of non-Abelian lattice gauge theories relevant to the standard model. This improvement results from our algorithm having polynomial scaling with the number of colors in the gauge theory, achieved by utilizing oracle constructions relying on the sparsity of physical operators, in contrast to the exponential scaling seen in state-of-the-art Trotter methods, which employ explicit mappings onto Pauli operators. Our work demonstrates that the use of advanced algorithmic techniques leads to dramatic reductions in the cost of simulating fundamental interactions, bringing it in step with resources required for first-principles quantum simulation of chemistry. Published by the American Physical Society 2024
Simulating an open quantum system on a digital quantum computer often involves the use of Trotter-Suzuki (TS) Product Formulas (PF) to approximate the system's time evolution. Precise estimates for the required number of Trotter steps (and hence the overall gate count) can be crucial for minimising the computational cost of these methods. Building on established theoretical guarantees, we derive analytic bounds for the First- and Second-Order Deterministic and Randomised TS-PF, directly relating the number of Trotter steps to the model parameters, evolution time and precision. These bounds enable concrete resource estimation for each method. We then present a computationally efficient classical algorithm that uses diamond norm estimates of individual Liouvillian terms and a binary search to significantly reduce the Trotter steps required for a target precision. Our numerical results on two prototypical models - an XX-Spin Chain with boundary driving and local dephasing, and a Transverse-Field Ising Model - show that the theoretical (analytic) bounds are often overly conservative, whereas the empirical (optimised) bounds yield a significantly smaller number of Trotter steps for the same precision. Among the methods investigated, the Second-Order Randomised TS-PF typically achieves the lowest resource demands, especially for larger systems. These findings emphasise the significance of empirical bounding strategies in achieving more resource-efficient simulations of Markovian open quantum systems.